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![2 Graphs and Groups: Preliminaries
vertex-set is V(G) and whose edge-set is E(G) − F. If S = {u} and F = {e},
we shall, for short, denote G − S and G − F by G − u and G − e, respectively.
If S is a subset of the vertices of G, then G[S] will denote the subgraph of G
induced by S, that is, the subgraph consisting of the vertices in S and all of the
edges joining pairs of vertices from S.
An important modification of the foregoing definition of a graph gives what
is called a directed graph, or digraph for short. In a digraph D = (V(D), A(D))
the set A = A(D) consists of ordered pairs of vertices from V = V(D) and its
elements are called arcs. Again, an arc (u, v) is sometimes denoted by uv when
it is clear from the context whether we are referring to an arc or an edge. The
arc uv is said to be incident to v and incident from u; the vertex u is said to be
adjacent to v whereas v is adjacent from u. The number of arcs incident from a
vertex v is called its out-degree, denoted by degout(v), while the number of arcs
incident to v is called its in-degree and is denoted by degin(v). A digraph is said
to be regular if all of its vertices have the same out-degree or, equivalently, the
same in-degree. Sometimes, when we need to emphasise the fact that a graph
is not directed, we say that it is undirected.
The number of vertices of a graph G or digraph D is called its order and is
generally denoted by n = n(G) or n = n(D), while the number of edges or
arcs is called its size and is denoted by m = m(G) or m = m(D).
A sequence of distinct vertices of a graph, v1, v2, . . . , vk+1, and edges e1, e2,
. . . , ek such that each edge ei = vivi+1 is called a path. If we allow v1 and
vk+1, and only those, to be the same vertex, then we get what is called a
cycle.
The length of a path or a cycle in G is the number of edges in the path or
cycle. A path of length k is denoted by Pk+1 while a cycle of length k is denoted
by Ck. The distance between two vertices u, v in a connected graph G, denoted
by d(u, v), is the length of the shortest path joining u and v. The diameter of G
is the maximum value attained by d(u, v) as u, v run over V(G), and the girth
is the length of the shortest cycle.
In these definitions, if we are dealing with a digraph and the ei = vivi+1
are arcs, then the path or cycle is called a directed path or directed cycle,
respectively.
Given a digraph D, the underlying graph of D is the graph obtained from
D by considering each pair in A(D) to be an unordered pair. Given a graph G,
the digraph
←
→
G is obtained from G by replacing each edge in E(G) by a pair
of oppositely directed arcs. This way, a graph can always be seen as a special
case of a digraph.
We adopt the usual convention of representing graphs and digraphs by draw-
ings in which each vertex is shown by a dot, each edge by a curve joining the](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-24-2048.jpg)
![1.2 Groups 3
corresponding pair of dots and each arc (u, v) by a curve with an arrowhead
pointing in the direction from u to v.
A number of definitions on graphs and digraphs will be given as they are
required. However, several standard graph theoretic terms will be used but not
defined in these chapters; these can be found in any of the references [257] or
[259].
1.2 Groups
A permutation group will be a pair (, Y) where Y is a finite set and is a
subgroup of the symmetric group SY, that is, the group of all permutations of
Y. The stabiliser of an element y ∈ Y under the action of is denoted by y
while the orbit of y is denoted by (y). The Orbit-Stabiliser Theorem states
that, for any element y ∈ Y,
|| = |(y)| · |y|.
If the elements of Y are all in one orbit, then (, Y) is said to be a transitive
permutation group and is said to act transitively on Y. The permutation group
is said to act regularly on Y if it acts transitively and the stabiliser of any
element of Y is trivial. By the Orbit-Stabiliser Theorem, this is equivalent to
saying that acts transitively on Y and || = |Y|. Also, acts regularly on Y
is equivalent to saying that, for any y1, y2 ∈ Y, there exists exactly one α ∈
such that α(y1) = y2.
One important regular action of a permutation group arises as follows. Let
be any group, let Y = and, for any α ∈ , let λα be the permutation
of Y defined by λα(β) = αβ. Let L() be the set of all permutations λα for
all α ∈ . Then (L(), Y) defines a permutation group acting regularly on Y.
This is called the left regular representation of the group on itself. One can
similarly consider the right regular representation of the group on itself, and
this is denoted by (R(), Y).
The following is an important generalisation of the previous definitions. If
is a group and H ≤ , let Y = /H be the set of left cosets of H in . For any
α ∈ , let λH
α be a permutation on Y defined by λH
α (βH) = αβH. Let LH()
be the set of all λH
α for all α ∈ . Then (LH(), Y) defines a permutation
group that reduces to the left regular representation of if H = {1}.
Two permutation groups (1, Y1), (2, Y2) are said to be equivalent, denoted
by (1, Y1) ≡ (2, Y2), if there exists a bijective isomorphism φ : 1 → 2
and a bijection f : Y1 → Y2 such that, for all y ∈ Y1 and for all α ∈ 1,
f(α(x)) = φ(α)(f(x)).](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-25-2048.jpg)
![1.3 Graphs and groups 7
is the infinite free group on two elements. The dihedral group of degree n is
denoted by Dn. It has order 2n and also has a presentation with two generators:
α, β|α2
= 1, βn
= 1, α−1
βα = β−1
.
Determining a group from a given presentation is not an easy problem. The
reader who doubts this can try to show that the presentations
α, β : αβ2
= β3
α, βα2
= α3
β
and
α, β, γ : α3
= β3
= γ 3
= 1, αγ = γ α−1
, αβα−1
= βγβ−1
both give the trivial group. We shall of course make a very simple use of stan-
dard group presentations where these difficulties do not arise. The book [159]
is a standard reference for advanced work on group presentations.
The reader is referred to [147, 222] for any terms and concepts on group
theory that are used but not defined in these chapters and, in particular, to [49,
62] for more information on permutation groups.
1.3 Graphs and groups
Let G, G be two graphs. A bijection α : V(G) → V(G ) is called an isomor-
phism if
{u, v} ∈ E(G) ⇔ {α(u), α(v)} ∈ E(G ).
The graphs G, G are, in this case, said to be isomorphic, and this is denoted by
G G . Similarly, if D, D are digraphs, then a bijection α : V(D) → V(D ) is
called an isomorphism if
(u, v) ∈ A(D) ⇔ (α(u), α(v)) ∈ A(D ),
and in this case the digraphs D, D are also said to be isomorphic, and again
this is denoted by D D .
If the two graphs, or digraphs, in this definition are the same, then α is said
to be an automorphism of G or of D. The set of automorphisms of a graph or a
digraph is a group under composition of functions, and it is denoted by Aut(G)
or Aut(D).
Note that an automorphism α of G is an element of SV(G), although it is its
induced action on E(G) that determines whether α is an automorphism. This
fact, although clear from the definition of automorphism, is worth emphasising
when beginning to study automorphisms of graphs.](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-29-2048.jpg)
![10 Graphs and Groups: Preliminaries
Figure 1.6. The eight nonidentical graphs of order 3
Counting nonisomorphic graphs involves consideration of group symme-
tries. For more on this the reader is referred to [103].
1.4 Edge-automorphisms and line-graphs
Although we shall be dealing mostly with Aut(G) and its realisation as the
permutation group (Aut(G), V(G)), let us briefly look at other related groups
associated with G. In this section we shall assume that G is a nontrivial graph,
that is, its edge-set is nonempty.
An edge-automorphism of a graph G is a bijection θ on E(G) such that two
edges e, f are adjacent in G if and only if θ(e), θ(f) are also adjacent in G. The
set of all edge-automorphisms of G is a group under composition of functions,
and it is denoted by Aut1(G).
The concept of edge-automorphisms can perhaps be best understood within
the context of line-graphs. The line-graph L(G) of a graph G is defined as
the graph whose vertex-set is E(G) and in which two vertices are adjacent if
and only if the corresponding edges are adjacent in G. An automorphism of
L(G) is clearly an edge-automorphism of G and (Aut1(G), E(G)) is equivalent
to (Aut(L(G), V(L(G))). In this section we shall give the exact relationship
between Aut1(G) and Aut(G), that is, between the automorphism groups of G
and L(G).
As we described earlier, any automorphism α of G naturally induces a bijec-
tion α̂ on E(G) defined by α̂(uv) = α(u)α(v). It is an important (and easy to](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-32-2048.jpg)
![12 Graphs and Groups: Preliminaries
Figure 1.7. Graphs with edge-isomorphisms not induced by isomorphisms
Before proceeding let us first extend the idea of edge-automorphisms on
the edge-set of a graph to that of edge-isomorphisms between edge-sets of
different graphs.
Let G, G be two nontrivial graphs. A bijection θ : E(G) → E(G ) is an
edge-isomorphism if
e, f adjacent in G ⇔ θ(e), θ(f) adjacent in G .
Two graphs are said to be edge-isomorphic if there is an edge-isomorphism
between their edge-sets.
The graphs W1, W2 in Figure 1.7 are edge-isomorphic, although they are
not isomorphic. That is, there is an edge-isomorphism between their edge-
sets that cannot be induced by an isomorphism between their vertex-sets. This
means that their line-graphs are isomorphic even though the two graphs are not
themselves isomorphic.
Also, each of the graphs W3, W4, W5 in the same figure has edge-
automorphisms that are not induced by automorphisms. That is, the group
Aut∗
(Wi) is a strict subgroup of Aut1(Wi). In other words, Aut(L(Wi)) is larger
than Aut(Wi).
The following theorem of Whitney [258] says that these are essentially the
only cases when edge-isomorphisms that are not induced by isomorphisms can
arise. We give the statement of the theorem without proof, which, although
not deep or difficult, would lengthen this introductory chapter without adding
significant new insights.
Theorem 1.3 (Whitney) Let G, G be connected graphs different from the five
graphs in Figure 1.7. Let θ : E(G) → E(G ) be an edge-isomorphism. Then θ
is induced by an isomorphism from G to G .](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-34-2048.jpg)
![1.5 A word on issues of computational complexity 13
Whitney’s Theorem and Theorem 1.2 together therefore give the following
corollary.
Corollary 1.4 Let G be a nontrivial graph. Then Aut1(G) = Aut∗(G) if and
only if both of these conditions hold:
(i) not both W1, W2 are components of G;
(ii) none of Wi, i = 3, 4, 5 is components of G.
Moreover, Aut1(G) Aut(G) (that is, Aut(L(G)) Aut(G)) if and only if (i)
and (ii) hold and G has at most one isolated vertex and K2 is not a component
of G.
1.5 A word on issues of computational complexity
Although in this book we shall not concern ourselves with issues of computa-
tional complexity, it is perhaps worthwhile to say a few words in this regard
here in order to put matters into a better perspective. A student reading the def-
initions of isomorphic graphs and automorphisms might think that it is an easy
matter to determine in general whether two given graphs are isomorphic or to
compute the automorphism group of a graph. In fact, this is far from being the
case, and these problems are very hard to crack in practice, at least as far as
present knowledge goes.
In general, one considers that an efficient algorithm exists for finding a solu-
tion to a problem (for example, finding a nontrivial automorphism of a given
graph) if there is a general algorithm such that the number of operations that
it takes to solve the problem is a polynomial function of the size of the input
(say, the number of vertices in the graph); one says that the algorithm solves
the problem in polynomial time.
Of course, several terms in the previous sentence need exact definitions, but
we shall here take an intuitive approach and refer the reader to [36] or [82] for
the exact details on computational complexity.
Those problems for which an efficient (polynomial-time) algorithm exists
form the class denoted by P (which stands for ‘polynomial’). However, there
are several problems for which it is not known whether an efficient algorithm
does exist. In order to tackle this question of computational intractibility, two
important ideas have been developed.
Firstly, the class NP (which stands for ‘nondeterministic polynomial’) is
defined. Roughly (again we refer the reader to the textbooks cited earlier for
the exact details) this class contains all of those problems for which, given a
candidate solution, one can verify in polynomial time that it is in fact a correct](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-35-2048.jpg)
![14 Graphs and Groups: Preliminaries
solution. For example, the problem of determining whether a graph has a non-
trivial automorphism is in NP, since, given such a permutation of the vertices,
it is easy to determine in polynomial time that it is an automorphism.
Now the main question in computational complexity is whether P = NP
(clearly P ⊆ NP), and to tackle this question another important idea is intro-
duced. Given two problems A and B, one says that A is (polynomially) reducible
to B if, given an algorithm for solving B, it can be transformed in polynomial
time into an algorithm for solving A. Reducibility therefore introduces a hier-
archy between problems for, if A is reducible to B, then, in a sense, A cannot
be more difficult to solve (computationally) than B. In particular, if there is an
efficient algorithm for solving B, then there is also an efficient algorithm for
solving A.
Now, the question of reducibility took on special significance by the discov-
ery that in the class NP there are problems, called NP-complete, to which any
other problem in NP is reducible. In other words, if an efficient algorithm can
be found for any NP-complete problem, then all problems in NP would have
an efficient algorithm to solve them, and P would be equal to NP.
Now, it is not known whether the problem of determining if two graphs are
isomorphic, which lies clearly in NP, is NP-complete. In fact, if it turns out
that P is not equal to NP, then there is evidence to suggest that the problem of
graph isomorphism might lie strictly between the classes P and NP.
What all this means in practice is that, as far as present knowledge goes, no
general algorithm can determine in a guaranteed reasonable time whether two
graphs are isomorphic, or whether a given graph has a nontrivial automorphism
(these two problems are closely related [126]). It is known that for special types
of graphs (for example, trees, planar graphs and graphs with bounded degree)
an efficient algorithm does exist.
Computer packages can also help one to solve these problems, certainly
more efficiently than an attempt ‘by hand’ for large graphs, although their time
performance is not guaranteed (by what we said earlier). For example, the soft-
ware package MathematicaTM has a combinatorics extension that, amongst
other things, finds graph automorphisms and isomorphisms. A more specialised
package, and one that is freely available from
www.combinatorialmath.org.ca/gg/index.html
is Groups Graphs [131], developed by Bill Kocay. This package contains
several combinatorial routines related to graphs, digraphs, combinatorial designs
and their automorphism groups and also embeddings of graphs on some sur-
faces and a graph isomorphism algorithm. It is easy to use, and it has a pleasant
graphical user interface. It is also very useful simply for drawing diagrams of](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-36-2048.jpg)
![1.6 Exercises 15
graphs. Although originally written for MacintoshTM computers, a version for
the unix-based Haiku operating system is in preparation, and this version will
contain several new features.
An important computer algebra package, which is also freely available, is
the system GAP [243]. This package performs very sophisticated routines in
discrete abstract algebra, in particular routines on permutation groups. It incor-
porates a number of extensions, one of which, GRAPE [235], deals specifically
with graphs, including their automorphisms and isomorphisms.
The computer package Sage [227] is an open-source competitor to systems
like MapleTM, MathematicaTM and MatlabTM. It incorporates several open-
source mathematical software like GAP and R, and it can be run via Sage-
MathCloud without the need of installing the system on one’s computer. It has
an excellent library of functions for doing graph theory. In this book we shall
present some constructions using GAP and Sage.
Finally, it should be mentioned that it is generally accepted that the best
package to tackle graph isomorphisms is nauty [181], developed by Brendan
McKay. In fact, the system GRAPE invokes nauty when computing automor-
phisms or isomorphisms.
1.6 Exercises
1.1 Draw all twenty nonidentical graphs with vertex-set {1, 2, 3, 4} that have three
edges. How many of them are nonisomorphic? In general, how many nonidentical
graphs on n vertices and m edges are there? How many are there on n vertices?
1.2 Let G be the graph in Figure 1.8. How many nonidentical labellings does G have
using the labels {1, 2, . . . , 6} on its vertices?
In general, how many nonidentical labellings does a graph G on n vertices
have using the labels {1, 2, . . . , n} on its vertices?
1.3 Show that if G is self-complementary (that is, G G), then n ≡ 0 mod 4 or
n ≡ 1 mod 4. Determine all self-complementary graphs on five vertices.
1.4 A well-known result due to Cayley says that the number of nonidentical trees on
n vertices is nn−2. Verify this for n = 4. Look up one of the several proofs of this
result.
Figure 1.8. How many distinct labellings does this graph have?](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-37-2048.jpg)
![16 Graphs and Groups: Preliminaries
The points 1, 2, . . . , n are drawn in a plane. A random tree is drawn joining
these points, with all possible spanning trees being equally likely. Let pn be the
probability that 1 is an endvertex of the tree. Show that limn→∞ pn = 1/e.
1.5 Find a graph with a pair of pseudosimilar edges.
1.6 Let Y = {1, 2, 3, 4} and let be the group acting on Y generated by the permu-
tation (1 2 3 4). Construct a digraph D whose vertex-set is Y and whose arc-set is
the orbit of the arc (1, 2) under the action of the permutation group (, Y × Y).
Is the whole of Aut(D)? Can a graph be obtained by taking the orbit of some
other arc or a union of orbits? Will always be the whole of Aut(D)?
1.7 Let P be a rectangular plate and Q a plate in the form of a rhombus. Show that
the groups of symmetry of P and Q are isomorphic as abstract groups but not
equivalent considered as permutation groups of the four vertices of P and Q.
Show that the same situation arises with the following two graphs: the cycle
on four vertices with an extra multiple edge and the complete graph K4 with an
edge deleted.
1.8 Show that the dihedral group Dn can be presented as
α, β|α2 = β2 = (αβ)n = 1 .
1.9 Let 1 be the abelian group defined by the presentation
α, β, γ |α3 = β3 = γ 3 = 1, [α, β] = [α, γ ] = [β, γ ] = 1 ,
where [α, β] = α−1β−1αβ is the commutator of α and β, and let 2 be the group
defined by the presentation
α, β, γ |α3 = β3 = γ 3 = 1, [β, α] = γ , [α, γ ] = [β, γ ] = 1 .
Show that although 1 and 2 are not isomorphic as abstract groups, and there-
fore the two permutation groups (L(1), 1) and (L(2), 2) are not equivalent,
still the cycle structures of the permutations of L(1) acting on 1 are the same
as those of the permutations of L(2) acting on 2.
Give an example of two permutation groups whose permutations have the
same cycle structures and which are isomorphic as abstract groups but are still
not equivalent as permutation groups.
1.10 Prove Theorem 1.1.
1.11 This exercise is intended to illustrate Theorem 1.1. Consider the action of the
alternating group A4 on the set X = {1, 2, 3, 4}. Let H be the stabiliser of the
element 1 under this action. Confirm that the left action of A4 on the left cosets
of H is equivalent to the action of A4 on X.
1.12 Let be a group of permutations acting on a set Y and let y, x be two elements of
Y that are in the same orbit under this action. Let α, β ∈ be two permutations
such that α(y) = β(y) = x. Prove that αy = βy.
Prove also that if x is in the same orbit as y and γ is a permutation such that
γ (y) = x and γ y = αy, then x = x.
1.13 Show that if the abelian permutation group acts transitively on Y, then its action
on Y is regular.
1.14 (a) Show that a 2-transitive permutation group is primitive.
(b) Show that if Aut(G), for a graph G, is 2-transitive, then either G or its com-
plement is a complete graph.](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-38-2048.jpg)
![1.7 Notes and guide to references 17
(c) Suppose that the transitive permutation group (, Y), with |Y| finite, is imprim-
itive. Show that the blocks of an imprimitivity block system have equal size.
1.15 Let G be a vertex-transitive graph whose automorphism group acts impri-
mitively on V(G). Show that the subgraphs of G induced by the blocks of an
imprimitivity block system are all isomorphic.
Suppose that each such subgraph is replaced by its complement, leaving the
other edges intact. Let G be the resulting graph. Show that Aut(G ) = Aut(G).
1.16 The Petersen graph can be defined as follows. Let N = {1, 2, 3, 4, 5}, and let the
vertices of the graph be all subsets of N of size 2 in which two vertices are adja-
cent if the corresponding subsets are disjoint. Use this definition and GAP (with
GRAPE) to construct the Petersen graph and to verify some of its properties.
1.7 Notes and guide to references
One of the standard texts on graph theory has, for many years, been [97]. More
recent books that give an excellent coverage of the subject are [28, 61, 257,
259]. The last reference is a short introduction that is quite sufficient back-
ground for this book. Biggs’ book [24] is the standard text on algebraic graph
theory, but the more recent [90] is also an excellent and up-to-date textbook on
the subject. The book [94] contains a number of recent and specialised survey
papers on various aspects of algebraic graph theory, particularly those dealing
with graph symmetries. A proof of Whitney’s Theorem can be found in [22].
We shall need only the most elementary notions of group theory. The text
[147] gives ample coverage for our purposes, while [222] provides a more
complete treatment. Two excellent books devoted entirely to permutation
groups are [49, 62]. Most of the results and definitions on permutation groups
that we have given here and others that we shall need can be found in the first
few chapters of these two books.
For a full discussion of the terms on computational complexity that were
introduced earlier rather intuitively, the reader is referred to the standard text-
book [82] or the more recent [36]. The book [126] and the references it cites
are suggested for those who are interested in the computational complexity of
the graph isomorphism problem. Those who are particularly interested in some
of the powerful algebraic techniques used to tackle this problem should look
at the papers [106, 155]. For practical computations on a computer with per-
mutation groups and graph automorphisms and isomorphisms in particular, the
systems [131, 181, 235, 243] are recommended.](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-39-2048.jpg)
![20 Various Types of Graph Symmetry
Now let a, b be in the same orbit, say V1. It then follows that a, b are not
adjacent. For suppose otherwise. Then the edge {u, v} is similar to the edge
{a, b} under the action of H; therefore the vertex v is similar to one of a or b
under this action, giving that v (which is in the orbit V2) is also in the orbit V1,
which contains a and b. But this is impossible since V1 ∩ V2 = ∅.
Hence, as required, we have that no two vertices from the same orbit can be
adjacent.
Corollary 2.2 If a graph G without isolated vertices is edge-transitive but not
vertex-transitive, then it is bipartite and the action of Aut(G) on V(G) has two
orbits that form the bipartition of V(G).
Proof Take H = Aut(G) in the previous theorem.
2.1.1 Semisymmetric graphs
The typical example of edge-transitive but not vertex-transitive graphs given
earlier is the complete bipartite graph with a different number of vertices in
the bipartition. These graphs are trivially not vertex-transitive because their
vertices have different degrees. Although regular graphs do exist that are edge-
transitive but not vertex-transitive, it is quite difficult to construct them. Such
graphs are now called semisymmetric graphs and they were first studied by
Folkman [77], who constructed the smallest possible semisymmetric graph
having twenty vertices.1 One construction of the Folkman Graph is described
in Exercise 2.5.
Here we shall describe another well-known semisymmetric graph, the Gray
Graph.2 For a long time nobody could find a smaller cubic semisymmetric
graph, and eventually it was formally proved in [160] that it is the smallest
cubic semisymmetric graph. We shall see a more systematic way of describing
it in subsequent chapters. Here we follow Bouwer’s construction in [33].
Consider a cycle on 54 vertices numbered consecutively from 0 to 53. To
form the Gray Graph G add the following edges to this cycle:
{1, 42}, {2, 15}, {3, 28}, {4, 33}, {5, 44}, {6, 53}, {7, 48}, {8, 21}, {9, 32},
{10, 45}, {11, 24}, {12, 41}, {13, 20}, {14, 31}, {16, 35}, {17, 40}, {18, 49},
{19, 0}, {22, 51}, {23, 30}, {25, 38}, {26, 43}, {27, 52}, {29, 36}, {34, 47},
1 More information about this graph, called the Folkman Graph, can be found on the MathWorld
page http://mathworld. wolfram.com/FolkmanGraph.html
2 The Gray Graph is also featured on MathWorld at http://mathworld.wolfram.com/Gray
Graph.html](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-42-2048.jpg)
![2.1 Transitivity 21
{37, 50}, {39, 46}.
It is tedious but not difficult to check that the permutations
α = (2 0 43)(3 53 43)(4 6 44)(7 45 33)(8 10 32)(11 31 21)
= (12 14 20)(15 19 41)(16 18 40)(22 24 30)(25 29 51)
= (26 28 52)(34 48 46)(35 49 39)(36 50 38)
and
β = (1 7 11 37 15 53 9 25 35)(2 6 10 38 16 0 8 24 36)
= (3 5 45 39 17 19 24 23 29)(4 44 46 40 18 20 22 30 28)
= (12 50 14 52 32 26 34 42 48)(13 51 31 27 33 43 47 41 49)
are automorphisms of G.
Note that the automorphism α fixes the vertex 1 and permutes cyclically its
neighbours 2, 42 and 0. Thus, in order to show that the graph is edge-transitive
it is sufficient to show that any odd-numbered vertex can be mapped into 1 by
an automorphism of G. This can be done by appropriate products of α and β.
For example, α4β maps vertex 53 to vertex 1.
However, the graph is not vertex-transitive because from an odd-numbered
vertex it is possible to have three different paths of length 4 joining the vertex
to some other common vertex (for example, vertex 1 to vertex 5), but this is
not possible from an even-numbered vertex.
Another way to show that the Gray Graph is not vertex-transitive is to con-
sider the distance sequences of its vertices [166]. The distance sequence of a
vertex v is the vector (a0, a1, . . . , ar) where ai is the number of vertices at dis-
tance i from v. In the case of the Gray Graph, the distance sequences of the
vertices in the two colour classes are (1, 3, 6, 12, 12, 12, 8) and (1, 3, 6, 12, 16,
12, 4), respectively, therefore these vertices cannot be in the same orbit under
the automorphism group of the graph.
Although the Sage package has the Gray Graph already implemented, it is
easy and instructive to show how to construct it following the specification
given earlier. First one creates the vertex-set which will be the list of numbers
from 1 to 54. Sage, like many computer languages such as Python, starts its
lists from 0. Therefore a list of length n produced by the command range(55)
would contain the numbers from 0 to 54. To start the list from 1 we have to
define the list of vertices as
vertices := range(1,55);](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-43-2048.jpg)
![22 Various Types of Graph Symmetry
The graph is then constructed first using the command DiGraph so that
we do not need to repeat every pair of adjacent vertices twice. This com-
mands basically takes two parameters. The first parameter is the vertex-set
of the graph to be constructed, and the second parameter is a boolean function
(defined with the lambda construct) of two variables which compares all pos-
sible pairs of the vertex-set and an edge is drawn between any pair of vertices
for which the function returns True.
dgray := DiGraph([vertices, lambda i, j:
(i == mod[j + 1, 54]) or
(i == 53 and j == 54) or
(i == 1 and j == 42) or
(i == 2 and j == 15) or
(i == 3 and j == 28) or
(i == 4 and j == 33) or
(i == 5 and j == 44) or
(i == 6 and j == 53) or
(i == 7 and j == 48) or
(i == 8 and j == 21) or
(i == 9 and j == 32) or
(i == 10 and j == 45) or
(i == 11 and j == 24) or
(i == 12 and j == 41) or
(i == 13 and j == 20) or
(i == 14 and j == 31) or
(i == 16 and j == 35) or
(i == 17 and j == 40) or
(i == 18 and j == 49) or
(i == 19 and j == 54 or
(i == 22 and j == 51) or
(i == 23 and j == 30) or
(i == 25 and j == 38) or
(i == 26 and j == 43) or
(i == 27 and j == 52) or
(i == 29 and j == 36) or
(i == 34 and j == 47) or
(i == 37 and j == 50) or
(i == 39 and j == 46) ] )
This digraph is then changed into an undirected graph with the following
command which changes every arc into an edge.](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-44-2048.jpg)
![2.2 Asymmetric graphs 27
Define a graph Hρ with vertex-set the pairs in Eρ and such that each pair
{v, w} is adjacent to the pair {ρ(v), ρ(w)}. In Hρ, all degrees are at most 2.
Degrees equal to 1 could arise because {ρ(v), ρ(w)} could contain an element
not in W, and so the pair would not be in Eρ. Degrees equal to 2 could arise
because {v, w} could be adjacent to both {ρ(v), ρ(w)} and {ρ−1(v), ρ−1(w)}.
Therefore the components of Hρ are isolated vertices, paths or cycles. Let Iρ
be a set of independent (that is, mutually not adjacent) vertices in Hρ. There-
fore, for any pair {v, w} ∈ Iρ, {ρ(v), ρ(w)} is not in Iρ.
Now, all isolated vertices in Hρ are independent, at least half of the vertices
on a path are independent and at least one third of the vertices on a cycle are
independent, the extreme case here being a triangle. Therefore
|Iρ| ≥ |Eρ|/3 ≥ 2g(t − 2)/6,
as required.
Corollary 2.6 Let G ∈ G(n, 1
2 ), W ⊂ V = V(G) and |W| = t. Let ρ : W → V
be an injective function that is not the identity. Let g = g(ρ) be the number of
elements w ∈ W such that ρ(w) = w. Let Sρ be the event
‘ρ gives an isomorphism from G[W] to G[ρ(W)]’.
Then
P(Sρ) ≤
1
2
2g(t−2)/6
.
Proof Let Iρ be the set constructed in the previous lemma. Now, for a given
pair {v, w} ∈ Iρ, the event
‘{v, w} and {ρ(v), ρ(w)} are both edges or nonedges’
has probability 1/2. These events, as they range over all pairs {v, w} ∈ Iρ, are
mutually independent because they involve distinct pairs. But Sρ requires all
these events simultaneously. Therefore, by independence,
P(Sρ) ≤
1
2
|Iρ|
≤
1
2
2g(t−2)/6
,
as required.
The result of this corollary is the crux of the proof of the following theo-
rem: There are too many independent correct ‘hits’ required for ρ to be an
isomorphism, and the probability therefore becomes small as n increases.](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-49-2048.jpg)
![28 Various Types of Graph Symmetry
Theorem 2.7 (Korshunov; Müller; Bollobás) Let k be a fixed nonnegative
integer and let G ∈ G(n, 1
2 ). Let pn denote the probability that
∃W ⊆ V(G) = V = {1, 2, . . . , n},
with |W| = n − k and such that
∃ρ : W → V, ρ = id, ρ is an isomorphism from G[W] to G[ρ(W)].
Then, limn→∞ pn = 0.
Hence, a.e. graph has property Ak.
Proof Pick a particular W ⊂ V with |W| = n − k. This can be done in
n
n−k
ways, and
n
n − k
=
n(n − 1) . . . (n − k + 1)
k!
nk
.
Let t = n − k. Let ρ : W → V be injective and not the identity, and let
g = g(ρ) be the number of vertices of W that are moved by ρ. Let Sρ be the
event defined in the previous corollary.
Now, for a given value of g between 1 and t, how many functions ρ are there
such that g(ρ) = g? Such a function is determined by the set {w : ρ(w) = w}
and by the values it takes on this set. Therefore, there are less than n2g such ρ.
Therefore, for a given fixed W, the probability of a nontrivial isomorphism is
given by
ρ=id
P(Sρ) =
t
g=1
ρ:g(ρ)=g
P(Sρ)
≤
t
g=1
n2g
1
2
2g(t−2)/6
=
t
g=1
n2
2(2−t)/3
g
t
g=1
41/3
n2
2−t/3
g
.](https://image.slidesharecdn.com/85668-250224195714-51cb9982/75/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-50-2048.jpg)
Topics in Graph Automorphisms and Reconstruction 2nd Edition Josef Lauri
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- 5. Topics in GraphAutomorphisms and Reconstruction 2nd Edition Josef Lauri Digital Instant Download Author(s): Josef Lauri, Raffaele Scapellato ISBN(s): 9781316610442, 1316610446 Edition: 2 File Details: PDF, 1.63 MB Year: 2016 Language: english
- 8. LONDON MATHEMATICAL SOCIETYLECTURE NOTE SERIES Managing Editor: Professor M. Reid, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at http://www.cambridge.org/mathematics 312 Foundations of computational mathematics, Minneapolis 2002, F. CUCKER et al. (eds) 313 Transcendental aspects of algebraic cycles, S. MÜLLER-STACH & C. PETERS (eds) 314 Spectral generalizations of line graphs, D. CVETKOVIĆ, P. ROWLINSON & S. SIMIĆ 315 Structured ring spectra, A. BAKER & B. RICHTER (eds) 316 Linear logic in computer science, T. EHRHARD, P. RUET, J.-Y. GIRARD & P. SCOTT (eds) 317 Advances in elliptic curve cryptography, I.F. BLAKE, G. SEROUSSI & N.P. SMART (eds) 318 Perturbation of the boundary in boundary-value problems of partial differential equations, D. HENRY 319 Double affine Hecke algebras, I. CHEREDNIK 320 L-functions and Galois representations, D. BURNS, K. BUZZARD & J. NEKOVÁŘ (eds) 321 Surveys in modern mathematics, V. PRASOLOV & Y. ILYASHENKO (eds) 322 Recent perspectives in random matrix theory and number theory, F. MEZZADRI & N.C. SNAITH (eds) 323 Poisson geometry, deformation quantisation and group representations, S. GUTT et al (eds) 324 Singularities and computer algebra, C. LOSSEN & G. PFISTER (eds) 325 Lectures on the Ricci flow, P. TOPPING 326 Modular representations of finite groups of Lie type, J.E. HUMPHREYS 327 Surveys in combinatorics 2005, B.S. WEBB (ed) 328 Fundamentals of hyperbolic manifolds, R. CANARY, D. EPSTEIN & A. MARDEN (eds) 329 Spaces of Kleinian groups, Y. MINSKY, M. SAKUMA & C. SERIES (eds) 330 Noncommutative localization in algebra and topology, A. RANICKI (ed) 331 Foundations of computational mathematics, Santander 2005, L. M PARDO, A. PINKUS, E. SÜLI & M.J. TODD (eds) 332 Handbook of tilting theory, L. ANGELERI HÜGEL, D. HAPPEL & H. KRAUSE (eds) 333 Synthetic differential geometry (2nd Edition), A. KOCK 334 The Navier–Stokes equations, N. RILEY & P. DRAZIN 335 Lectures on the combinatorics of free probability, A. NICA & R. SPEICHER 336 Integral closure of ideals, rings, and modules, I. SWANSON & C. HUNEKE 337 Methods in Banach space theory, J.M.F. CASTILLO & W.B. JOHNSON (eds) 338 Surveys in geometry and number theory, N. YOUNG (ed) 339 Groups St Andrews 2005 I, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 340 Groups St Andrews 2005 II, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 341 Ranks of elliptic curves and random matrix theory, J.B. CONREY, D.W. FARMER, F. MEZZADRI & N.C. SNAITH (eds) 342 Elliptic cohomology, H.R. MILLER & D.C. RAVENEL (eds) 343 Algebraic cycles and motives I, J. NAGEL & C. PETERS (eds) 344 Algebraic cycles and motives II, J. NAGEL & C. PETERS (eds) 345 Algebraic and analytic geometry, A. NEEMAN 346 Surveys in combinatorics, A. HILTON & J. TALBOT (eds) 347 Surveys in contemporary mathematics, N. YOUNG & Y. CHOI (eds) 348 Transcendental dynamics and complex analysis, P.J. RIPPON & G.M. STALLARD (eds) 349 Model theory with applications to algebra and analysis I, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 350 Model theory with applications to algebra and analysis II, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 351 Finite von Neumann algebras and masas, A.M. SINCLAIR & R.R. SMITH 352 Number theory and polynomials, J. MCKEE & C. SMYTH (eds) 353 Trends in stochastic analysis, J. BLATH, P. MÖRTERS & M. SCHEUTZOW (eds) 354 Groups and analysis, K. TENT (ed) 355 Non-equilibrium statistical mechanics and turbulence, J. CARDY, G. FALKOVICH & K. GAWEDZKI 356 Elliptic curves and big Galois representations, D. DELBOURGO 357 Algebraic theory of differential equations, M.A. H. MACCALLUM & A.V. MIKHAILOV (eds) 358 Geometric and cohomological methods in group theory, M.R. BRIDSON, P.H. KROPHOLLER & I.J. LEARY (eds) 359 Moduli spaces and vector bundles, L. BRAMBILA-PAZ, S.B. BRADLOW, O. GARCÍA-PRADA & S. RAMANAN (eds) 360 Zariski geometries, B. ZILBER 361 Words: notes on verbal width in groups, D. SEGAL 362 Differential tensor algebras and their module categories, R. BAUTISTA, L. SALMERÓN & R. ZUAZUA 363 Foundations of computational mathematics, Hong Kong 2008, F. CUCKER, A. PINKUS & M.J. TODD (eds) 364 Partial differential equations and fluid mechanics, J.C. ROBINSON & J.L. RODRIGO (eds) 365 Surveys in combinatorics 2009, S. HUCZYNSKA, J.D. MITCHELL & C.M. RONEY-DOUGAL (eds) 366 Highly oscillatory problems, B. ENGQUIST, A. FOKAS, E. HAIRER & A. ISERLES (eds) 367 Random matrices: high dimensional phenomena, G. BLOWER 368 Geometry of Riemann surfaces, F.P. GARDINER, G. GONZÁLEZ-DIEZ & C. KOUROUNIOTIS (eds) 369 Epidemics and rumours in complex networks, M. DRAIEF & L. MASSOULIÉ 370 Theory of p-adic distributions, S. ALBEVERIO, A.YU. KHRENNIKOV & V.M. SHELKOVICH 371 Conformal fractals, F. PRZYTYCKI & M. URBAŃSKI 372 Moonshine: the first quarter century and beyond, J. LEPOWSKY, J. MCKAY & M.P. TUITE (eds)
- 9. 373 Smoothness, regularityand complete intersection, J. MAJADAS & A.G. RODICIO 374 Geometric analysis of hyperbolic differential equations: an introduction, S. ALINHAC 375 Triangulated categories, T. HOLM, P. JØRGENSEN & R. ROUQUIER (eds) 376 Permutation patterns, S. LINTON, N. RUŠKUC & V. VATTER (eds) 377 An introduction to Galois cohomology and its applications, G. BERHUY 378 Probability and mathematical genetics, N.H. BINGHAM & C. M. GOLDIE (eds) 379 Finite and algorithmic model theory, J. ESPARZA, C. MICHAUX & C. STEINHORN (eds) 380 Real and complex singularities, M. MANOEL, M.C. ROMERO FUSTER & C.T. C WALL (eds) 381 Symmetries and integrability of difference equations, D. LEVI, P. OLVER, Z. THOMOVA & P. WINTERNITZ (eds) 382 Forcing with random variables and proof complexity, J. KRAJÍČEK 383 Motivic integration and its interactions with model theory and non-Archimedean geometry I, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 384 Motivic integration and its interactions with model theory and non-Archimedean geometry II, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 385 Entropy of hidden Markov processes and connections to dynamical systems, B. MARCUS, K. PETERSEN & T. WEISSMAN (eds) 386 Independence-friendly logic, A.L. MANN, G. SANDU & M. SEVENSTER 387 Groups St Andrews 2009 in Bath I, C.M. CAMPBELL et al. (eds) 388 Groups St Andrews 2009 in Bath II, C.M. CAMPBELL et al. (eds) 389 Random fields on the sphere, D. MARINUCCI & G. PECCATI 390 Localization in periodic potentials, D.E. PELINOVSKY 391 Fusion systems in algebra and topology, M. ASCHBACHER, R. KESSAR & B. OLIVER 392 Surveys in combinatorics 2011, R. CHAPMAN (ed) 393 Non-abelian fundamental groups and Iwasawa theory, J. COATES et al. (eds) 394 Variational problems in differential geometry, R. BIELAWSKI, K. HOUSTON & M. SPEIGHT (eds) 395 How groups grow, A. MANN 396 Arithmetic differential operators over the p-adic integers, C.C. RALPH & S.R. SIMANCA 397 Hyperbolic geometry and applications in quantum chaos and cosmology, J. BOLTE & F. STEINER (eds) 398 Mathematical models in contact mechanics, M. SOFONEA & A. MATEI 399 Circuit double cover of graphs, C.-Q. ZHANG 400 Dense sphere packings: a blueprint for formal proofs, T. HALES 401 A double Hall algebra approach to affine quantum Schur–Weyl theory, B. DENG, J. DU & Q. FU 402 Mathematical aspects of fluid mechanics, J.C. ROBINSON, J. L. RODRIGO & W. SADOWSKI (eds) 403 Foundations of computational mathematics, Budapest 2011, F. CUCKER, T. KRICK, A. PINKUS & A. SZANTO (eds) 404 Operator methods for boundary value problems, S. HASSI, H.S. V. DE SNOO & F.H. SZAFRANIEC (eds) 405 Torsors, étale homotopy and applications to rational points, A.N. SKOROBOGATOV (ed) 406 Appalachian set theory, J. CUMMINGS & E. SCHIMMERLING (eds) 407 The maximal subgroups of the low-dimensional finite classical groups, J.N. BRAY, D.F. HOLT & C.M. RONEY-DOUGAL 408 Complexity science: the Warwick master’s course, R. BALL, V. KOLOKOLTSOV & R.S. MACKAY (eds) 409 Surveys in combinatorics 2013, S.R. BLACKBURN, S. GERKE & M. WILDON (eds) 410 Representation theory and harmonic analysis of wreath products of finite groups, T. CECCHERINI-SILBERSTEIN, F. SCARABOTTI & F. TOLLI 411 Moduli spaces, L. BRAMBILA-PAZ, O. GARCÍA-PRADA, P. NEWSTEAD & R.P. THOMAS (eds) 412 Automorphisms and equivalence relations in topological dynamics, D.B. ELLIS & R. ELLIS 413 Optimal transportation, Y. OLLIVIER, H. PAJOT & C. VILLANI (eds) 414 Automorphic forms and Galois representations I, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 415 Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 416 Reversibility in dynamics and group theory, A.G. O’FARRELL & I. SHORT 417 Recent advances in algebraic geometry, C.D. HACON, M. MUSTAŢĂ & M. POPA (eds) 418 The Bloch–Kato conjecture for the Riemann zeta function, J. COATES, A. RAGHURAM, A. SAIKIA & R. SUJATHA (eds) 419 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations, J.C. MEYER & D.J. NEEDHAM 420 Arithmetic and geometry, L. DIEULEFAIT et al (eds) 421 O-minimality and Diophantine geometry, G.O. JONES & A.J. WILKIE (eds) 422 Groups St Andrews 2013, C.M. CAMPBELL et al (eds) 423 Inequalities for graph eigenvalues, Z. STANIĆ 424 Surveys in combinatorics 2015, A. CZUMAJ et al. (eds) 425 Geometry, topology and dynamics in negative curvature, C.S. ARAVINDA, F.T. FARRELL & J.-F. LAFONT (eds) 426 Lectures on the theory of water waves, T. BRIDGES, M. GROVES & D. NICHOLLS (eds) 427 Recent advances in Hodge theory, M. KERR & G. PEARLSTEIN (eds) 428 Geometry in a Fréchet context, C.T.J. DODSON, G. GALANIS & E. VASSILIOU 429 Sheaves and functions modulo p, L. TAELMAN 430 Recent progress in the theory of the Euler and NavierStokes equations, J.C. ROBINSON, J.L. RODRIGO, W. SADOWSKI & A. VIDAL-LÓPEZ (eds) 431 Harmonic and subharmonic function theory on the real hyperbolic ball, M. STOLL 432 Topics in graph automorphisms and reconstruction (2nd Edition), J. LAURI & R. SCAPELLATO 433 Regular and irregular holonomic D-modules, M. KASHIWARA & P. SCHAPIRA 434 Analytic semigroups and semilinear initial boundary value problems (2nd Edition), K. TAIRA 435 Graded rings and graded Grothendieck groups, R. HAZRAT
- 10. London Mathematical SocietyLecture Note Series: 432 Topics in Graph Automorphisms and Reconstruction Second Edition JOSEF LAURI University of Malta RAFFAELE SCAPELLATO Politecnico di Milano
- 11. University Printing House,Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781316610442 c Josef Lauri and Raffaele Scapellato 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library. Library of Congress Cataloguing-in-Publication Data Names: Lauri, Josef, 1955– | Scapellato, Raffaele, 1955– Title: Topics in graph automorphisms and reconstruction / Josef Lauri and Raffaele Scapellato. Description: 2nd edition. | Cambridge : Cambridge University Press, 2016. | Series: London Mathematical Society lecture note series; 432 | Includes bibliographical references and index. Identifiers: LCCN 2016014849 | ISBN 9781316610442 (pbk. : alk. paper) Subjects: LCSH: Graph theory. | Automorphisms. | Reconstruction (Graph theory) Classification: LCC QA166.L39 2016 | DDC 511/.5–dc23 LC record available at https://lccn.loc.gov/2016014849 ISBN 978-1-316-61044-2 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 12. Lil Mary Anne, Christina, Beppeu Sandrina A Fiorella
- 13. London Mathematical SocietyLecture Note Series: 432 Topics in Graph Automorphisms and Reconstruction JOSEF LAURI University of Malta RAFFAELE SCAPELLATO Politecnico di Milano
- 15. Contents Preface to theSecond Edition page xi Preface to the First Edition xiii 1 Graphs and Groups: Preliminaries 1 1.1 Graphs and digraphs 1 1.2 Groups 3 1.3 Graphs and groups 7 1.4 Edge-automorphisms and line-graphs 10 1.5 A word on issues of computational complexity 13 1.6 Exercises 15 1.7 Notes and guide to references 17 2 Various Types of Graph Symmetry 18 2.1 Transitivity 18 2.2 Asymmetric graphs 25 2.3 Graph symmetries and the spectrum 29 2.4 Simple eigenvalues 31 2.5 Higher symmetry conditions 32 2.6 Exercises 35 2.7 Notes and guide to references 38 3 Cayley Graphs 39 3.1 Cayley colour graphs 39 3.2 Frucht’s and Bouwer’s Theorems 42 3.3 Cayley graphs and digraphs 44 3.4 The Doyle-Holt Graph 47 3.5 Non-Cayley vertex-transitive graphs 48 3.6 Coset graphs and Sabidussi’s Theorem 49 3.7 Double coset graphs and semisymmetric graphs 51 vii
- 16. viii Contents 3.8 Hamiltonicity53 3.9 Characters of abelian groups and Cayley graphs 55 3.10 Growth rates 56 3.11 Exercises 58 3.12 Notes and guide to references 62 4 Orbital Graphs and Strongly Regular Graphs 64 4.1 Definitions and basic properties 64 4.2 Rank 3 groups 68 4.3 Strongly regular graphs 69 4.4 The Integrality Condition 70 4.5 Moore graphs 73 4.6 Exercises 75 4.7 Notes and guide to references 77 5 Graphical Regular Representations and Pseudosimilarity 79 5.1 Elementary results 79 5.2 Abelian groups 80 5.3 Pseudosimilarity 81 5.4 Some basic results 82 5.5 Several pairs of pseudosimilar vertices 84 5.6 Several pairs of pseudosimilar edges 85 5.7 Large sets of mutually pseudosimilar vertices 86 5.8 Exercises 88 5.9 Notes and guide to references 91 6 Products of Graphs 92 6.1 General products of graphs 93 6.2 Direct product 95 6.3 Cartesian product 97 6.4 More products 99 6.5 Stability and two-fold automorphisms 102 6.6 Additional remarks on graph products 105 6.7 Exercises 105 6.8 Notes and guide to references 108 7 Special Classes of Vertex-Transitive Graphs and Digraphs 109 7.1 Generalised Petersen graphs 110 7.2 Kneser graphs and odd graphs 114 7.3 Metacirculant graphs 115 7.4 The quasi-Cayley graphs and digraphs 117 7.5 Generalised Cayley graphs 119
- 17. Contents ix 7.6 Exercises120 7.7 Notes and guide to references 122 8 The Reconstruction Conjectures 123 8.1 Definitions 124 8.2 Some basic results 126 8.3 Maximal planar graphs 132 8.4 Digraphs and degree-associated reconstruction 136 8.5 Exercises 138 8.6 Notes and guide to references 139 9 Reconstructing from Subdecks 140 9.1 The endvertex-deck 140 9.2 Reconstruction numbers 141 9.3 The characteristic polynomial deck 144 9.4 Exercises 147 9.5 Notes and guide to references 147 10 Counting Arguments in Vertex-Reconstruction 149 10.1 Kocay’s Lemma 149 10.2 Counting spanning subgraphs 151 10.3 The characteristic and the chromatic polynomials 154 10.4 Exercises 155 10.5 Notes and guide to references 156 11 Counting Arguments in Edge-Reconstruction 157 11.1 Definitions and notation 157 11.2 Homomorphisms of structures 159 11.3 Lovász’ and Nash-Williams’ Theorems 163 11.4 Extensions 166 11.5 Exercises 168 11.6 Notes and guide to references 169 References 171 List of Notations 185 Index of Terms and Definitions 187
- 19. Preface to theSecond Edition In this second edition of our book we have tried to maintain the same structure as the first edition, namely a text which, although not providing an exhaustive coverage of graph symmetries and reconstruction, provides a detailed cover- age of some particular areas (generally motivated by our own research inter- est), which is not a haphazard collection of results but which presents a clear pathway through this thick forest. And our aim remains that of producing a text which can relatively quickly guide the reader to the point of being able to understand and carry out research in the topics which we cover. Among the additions in this edition we point out the use of the free com- puter programs GAP, GRAPE and Sage to construct and investigate some well- known graphs, including examples with properties like being semisymmetric, a topic which was treated in the first edition but for which examples are not easy to construct ‘by hand’. We have also updated some chapters with new results, improved the presentation and proofs of others, and introduced short treatments of topics such as character theory of abelian groups and their Cay- ley graphs to emphasise the connection between graph theory and other areas of mathematics. We have corrected a number of errors which we found in the first edition, and for this we would like to thank colleagues who have pointed out several of them, particularly Bill Kocay, Virgilio Pannone and Alex Scott. A special thanks goes to Russell Mizzi for help with overhauling Chapter 6, where we also introduce the new idea of two-fold isomorphisms, and to Leonard Soicher and Matan Zif-Av for several helpful tips regarding the use of GAP and GRAPE. The second author would like to thank the Politecnico di Milano for giving him the opportunity, by means of a sabbatical, to focus on the work needed xi
- 20. xii Preface tothe Second Edition to complete the current edition of this book. He also thanks the University of Malta for its kind hospitality during this sabbatical. The authors will maintain a list of corrections and addenda at http://staff.um .edu.mt/josef.lauri. Josef Lauri Raffaele Scapellato
- 21. Preface to theFirst Edition This book arose out of lectures given by the first author to Masters students at the University of Malta and by the second author at the Università Cattolica di Brescia. This book is not intended to be an exhaustive coverage of graph theory. There are many excellent texts that do this, some of which are mentioned in the References. Rather, the intention is to provide the reader with a more in-depth coverage of some particular areas of graph theory. The choice of these areas has been largely governed by the research interests of the authors, and the flavour of the topics covered is predominantly algebraic, with emphasis on symmetry properties of graphs. Thus, standard topics such as the automorphism group of a graph, Frucht’s Theorem, Cayley graphs and coset graphs, and orbital graphs are presented early on because they provide the background for most of the work presented in later chapters. Here, more specialised topics are tackled, such as graphical regular representations, pseudosimilarity, graph products, Hamiltonicity of Cayley graphs and special types of vertex-transitive graphs, including a brief treatment of the difficult topic of classifying vertex-transitive graphs. The last four chapters are devoted to the Reconstruction Problem, and even here greater emphasis is given to those results that are of a more algebraic character and involve the symmetry of graphs. A special chapter is devoted to graph products. Such operations are often used to provide new examples from existing ones but are seldom studied for their intrinsic value. Throughout we have tried to present results and proofs, many of which are not usually found in textbooks but have to be looked for in journal papers. Also, we have tried, where possible, to give a treatment of some of these topics that is different from the standard published material (for example, the chapter on graph products and much of the work on reconstruction). xiii
- 22. xiv Preface tothe First Edition Although the prerequisites for reading this book are quite modest (exposure to a first course in graph theory and some discrete mathematics, and elemen- tary knowledge about permutation groups and some linear algebra), it was our intention when preparing this book that a student who has mastered its con- tents would be in a good position to understand the current state of research in most of the specialised topics covered, would be able to read with profit journal papers in these areas, and would hopefully have his or her interest sufficiently aroused to consider carrying out research in one of these areas of graph theory. We would finally like to thank Professor Caroline Series for showing an interest in this book when it was still in an early draft form and the staff at Cambridge University Press for their help and encouragement, especially Roger Astley, Senior Editor, Mathematical Sciences, and, for technical help with L ATEX, Alison Woollatt, who, with a short style file, solved problems that would have baffled us for ages. Thanks are also due to Elise Oranges, who edited this book thoroughly and pointed out several corrections. The first author would also like to thank the Academic Work Resources Fund Committee and the Computing Services Centre of the University of Malta, the first for some financial help while writing this book and the second for technical assistance. He also thanks his M.Sc. students at the University of Malta, who worked through draft chapters of this book and whose comments and criticism helped to improve the final product. Josef Lauri Raffaele Scapellato
- 23. 1 Graphs and Groups:Preliminaries 1.1 Graphs and digraphs In these chapters a graph G = (V(G), E(G)) will consist of two disjoint sets: a nonempty set V = V(G) whose elements will be called vertices and a set E = E(G) whose elements, called edges, will be unordered pairs of distinct elements of V. Unless explicitly stated otherwise, the set of vertices will always be finite. An edge, {u, v}, u, v ∈ V, is also denoted by uv. Sometimes E is allowed to be a multiset, that is, the same edge can be repeated more than once in E. Such edges are called multiple edges. Also, edges uu consisting of a pair of repeated vertices are sometimes allowed; such edges are called loops. But unless otherwise stated, it will always be assumed that a graph does not have loops or multiple edges. The complement of the graph G, denoted by G, has the same vertex-set as G, but two distinct vertices are adjacent in the complement if and only if they are not adjacent in G. The degree of a vertex v, denoted by deg(v), is the number of edges in E(G) to which v belongs. A vertex of degree k is sometimes said to be a k-vertex. Two vertices belonging to the same edge are said to be adjacent, while a vertex and an edge to which it belongs are said to be incident. A loop incident to a vertex v contributes a value of 2 to deg(v). A graph is said to be regular if all of its vertices have the same degree. A regular graph with degree equal to 3 is sometimes called cubic. The minimum and maximum degrees of G are denoted by δ = δ(G) and = (G), respectively. In general, given any two sets A, B, then A−B will denote their set-theoretical difference, that is, the set consisting of all of the elements that are in A but not in B. Also, a set containing k elements is often said to be a k-set. If S is a set of vertices of a graph G, then G−S will denote the graph obtained by removing S from V(G) and removing from E(G) all edges incident to some vertex in S. If F is a set of edges of G, then G − F will denote the graph whose 1
- 24. 2 Graphs andGroups: Preliminaries vertex-set is V(G) and whose edge-set is E(G) − F. If S = {u} and F = {e}, we shall, for short, denote G − S and G − F by G − u and G − e, respectively. If S is a subset of the vertices of G, then G[S] will denote the subgraph of G induced by S, that is, the subgraph consisting of the vertices in S and all of the edges joining pairs of vertices from S. An important modification of the foregoing definition of a graph gives what is called a directed graph, or digraph for short. In a digraph D = (V(D), A(D)) the set A = A(D) consists of ordered pairs of vertices from V = V(D) and its elements are called arcs. Again, an arc (u, v) is sometimes denoted by uv when it is clear from the context whether we are referring to an arc or an edge. The arc uv is said to be incident to v and incident from u; the vertex u is said to be adjacent to v whereas v is adjacent from u. The number of arcs incident from a vertex v is called its out-degree, denoted by degout(v), while the number of arcs incident to v is called its in-degree and is denoted by degin(v). A digraph is said to be regular if all of its vertices have the same out-degree or, equivalently, the same in-degree. Sometimes, when we need to emphasise the fact that a graph is not directed, we say that it is undirected. The number of vertices of a graph G or digraph D is called its order and is generally denoted by n = n(G) or n = n(D), while the number of edges or arcs is called its size and is denoted by m = m(G) or m = m(D). A sequence of distinct vertices of a graph, v1, v2, . . . , vk+1, and edges e1, e2, . . . , ek such that each edge ei = vivi+1 is called a path. If we allow v1 and vk+1, and only those, to be the same vertex, then we get what is called a cycle. The length of a path or a cycle in G is the number of edges in the path or cycle. A path of length k is denoted by Pk+1 while a cycle of length k is denoted by Ck. The distance between two vertices u, v in a connected graph G, denoted by d(u, v), is the length of the shortest path joining u and v. The diameter of G is the maximum value attained by d(u, v) as u, v run over V(G), and the girth is the length of the shortest cycle. In these definitions, if we are dealing with a digraph and the ei = vivi+1 are arcs, then the path or cycle is called a directed path or directed cycle, respectively. Given a digraph D, the underlying graph of D is the graph obtained from D by considering each pair in A(D) to be an unordered pair. Given a graph G, the digraph ← → G is obtained from G by replacing each edge in E(G) by a pair of oppositely directed arcs. This way, a graph can always be seen as a special case of a digraph. We adopt the usual convention of representing graphs and digraphs by draw- ings in which each vertex is shown by a dot, each edge by a curve joining the
- 25. 1.2 Groups 3 correspondingpair of dots and each arc (u, v) by a curve with an arrowhead pointing in the direction from u to v. A number of definitions on graphs and digraphs will be given as they are required. However, several standard graph theoretic terms will be used but not defined in these chapters; these can be found in any of the references [257] or [259]. 1.2 Groups A permutation group will be a pair (, Y) where Y is a finite set and is a subgroup of the symmetric group SY, that is, the group of all permutations of Y. The stabiliser of an element y ∈ Y under the action of is denoted by y while the orbit of y is denoted by (y). The Orbit-Stabiliser Theorem states that, for any element y ∈ Y, || = |(y)| · |y|. If the elements of Y are all in one orbit, then (, Y) is said to be a transitive permutation group and is said to act transitively on Y. The permutation group is said to act regularly on Y if it acts transitively and the stabiliser of any element of Y is trivial. By the Orbit-Stabiliser Theorem, this is equivalent to saying that acts transitively on Y and || = |Y|. Also, acts regularly on Y is equivalent to saying that, for any y1, y2 ∈ Y, there exists exactly one α ∈ such that α(y1) = y2. One important regular action of a permutation group arises as follows. Let be any group, let Y = and, for any α ∈ , let λα be the permutation of Y defined by λα(β) = αβ. Let L() be the set of all permutations λα for all α ∈ . Then (L(), Y) defines a permutation group acting regularly on Y. This is called the left regular representation of the group on itself. One can similarly consider the right regular representation of the group on itself, and this is denoted by (R(), Y). The following is an important generalisation of the previous definitions. If is a group and H ≤ , let Y = /H be the set of left cosets of H in . For any α ∈ , let λH α be a permutation on Y defined by λH α (βH) = αβH. Let LH() be the set of all λH α for all α ∈ . Then (LH(), Y) defines a permutation group that reduces to the left regular representation of if H = {1}. Two permutation groups (1, Y1), (2, Y2) are said to be equivalent, denoted by (1, Y1) ≡ (2, Y2), if there exists a bijective isomorphism φ : 1 → 2 and a bijection f : Y1 → Y2 such that, for all y ∈ Y1 and for all α ∈ 1, f(α(x)) = φ(α)(f(x)).
- 26. 4 Graphs andGroups: Preliminaries Figure 1.1. Aut(G), Aut(H) are isomorphic but not equivalent In this case we also say that the action of 1 on Y1 is equivalent to the action of 2 on Y2, and sometimes we denote this simply by 1 = 2, when the two sets on which the groups are acting is clear from the context. Figure 1.1 shows a simple example of two graphs whose automorphism groups (to be defined later in this chapter) are isomorphic as abstract groups but clearly not equivalent as permutation groups since the sets (of vertices) on which they act are not equal. (See also Exercise 1.7.) Note in particular that, if (1, Y1) ≡ (2, Y2), then apart from 1 2 as abstract groups, and |Y1| = |Y2|, the cycle structure of the permutations of 1 on Y1 must be the same as those of 2 on Y2. However, the converse is not true; that is, 1 and 2 could be isomorphic and the cycle structures of their respective actions could be the same, but (1, Y1) might not be equivalent to (2, Y2) (see Exercise 1.9). If (, Y) is a permutation group acting on Y and Y is a union of orbits of Y, then we can talk about the action of restricted to Y , that is, the permutation group (, Y ) where, for α ∈ and y ∈ Y , α(y ) is the same as in (, Y). When Y is a union of orbits we also say that it is invariant under the action of because in this case α(y ) ∈ Y for all α ∈ and y ∈ Y . Also, ( , Y ) is said to be a subpermutation group of (, Y) if ≤ and Y is a union of orbits of acting on Y. The following is a useful well-known result on permutation groups whose proof is not difficult and is left as an exercise (see Exercise 1.10). Theorem 1.1 Let (, Y) be a permutation group acting transitively on Y. Let y ∈ Y, let H = y be the stabiliser of y and let W be /H, the set of left cosets of H in . Then (, Y) is equivalent to (LH(), W). If (, Y) is not transitive, and O is the orbit containing y, then (LH(), W) is equivalent to the action of on Y restricted to O. In the context of groups and graphs we shall need the very important idea of a group acting on pairs of elements of a set. Thus, let (, Y) be a permutation
- 27. 1.2 Groups 5 groupacting on the set Y. By (, Y × Y) we shall mean the action on ordered pairs of Y induced by as follows: If α ∈ and x, y ∈ Y, then α((x, y)) = (α(x), α(y)). Similarly, by (, Y 2 ) we shall mean the action on unordered pairs of distinct elements of Y induced by α({x, y}) = {α(x), α(y)}. These ideas will be developed further in a later chapter. In later chapters we shall also need the notions of k-transitivity and primitiv- ity of a permutation group. In order to study permutation groups in more detail one has to dig deeper into the concept of transitivity. Suppose, for example, that Y is the set {1, 2, 3, 4, 5} and is the group generated by the permutation α = (1 2 3 4 5). Then clearly the permutation group (, Y) is transitive because for any i, j ∈ Y there is some power of α which maps i into j. But there is no power of α which, say, simultaneously maps 1 into 5 and 2 into 3. That is, not every ordered pair of distinct elements of Y can be mapped by a permutation in into any other given ordered pair of distinct elements. We therefore say that the permutation group (G, Y) is not 2-transitive. More generally, a permutation group (, Y) is said to be k-transitive if, given any two k-tuples (x1, x2, . . . , xk) and (y1, y2, . . . , yk) of distinct elements of Y, then there is an α ∈ such that (α(x1), α(x2), . . . , α(xk)) = (y1, y2, . . . , yk). Thus, a transitive permutation group is 1-transitive. Also, (, Y) is said to be k-homogeneous if, for any two k-subsets A, B of Y, there is an α ∈ such that α(A) = B, where α(A) = {α(a) : a ∈ A}. Finally, let (, Y) be transitive and suppose that R is an equivalence relation on Y, and let the equivalence classes of Y under R be Y1, Y2, . . . , Yr. Then (, Y) is said to be compatible with R if, for any α ∈ and any equiv- alence class Yi, the set α(Yi) is also an equivalence class. For example, if Y = {1, 2, 3, 4} and is the group generated by the permutation (1 2 3 4), then (, Y) is compatible with the relation whose equivalence classes are {1, 3} and {2, 4}. Any permutation group is clearly compatible with the trivial equivalence relations on Y, namely, those in which either all of Y is an equivalence class or when each singleton set is an equivalence class. If these are the only equiva- lence relations with which (, Y) is compatible, then the permutation group is said to be primitive. Otherwise it is imprimitive.
- 28. 6 Graphs andGroups: Preliminaries If (, Y) is imprimitive and R is a nontrivial equivalence relation on Y with which the permutation group is compatible, then the equivalence classes of R are called imprimitivity blocks and their set Y/R is an imprimitivity block system for the permutation group (, Y). It is an easy exercise (see Exercise 1.14) to show that a 2-transitive permu- tation group is primitive. We shall also need some elementary ideas on the presentation of a group in terms of generators and relations. Let be a group and let X ⊆ . A word in X is a product of a finite number of terms, each of which is an element of X or an inverse of an element of X. The set X is said to generate if every element in can be written as a word in X; in this case the elements of X are said to be generators of . A relation in X is an equality between two words in X. By taking inverses, any relation can be written in the form w = 1, where w is some word in X. If X generates and every relation in can be deduced from one of the relations w1 = 1, w2 = 1, . . . in X, then we write = X|w1 = 1, w2 = 1, . . . . This is called a presentation of in terms of generators and relations. The group is said to be finitely generated (respectively, finitely related) if |X| (respectively, the number of relations) is finite; it is called finitely presented, or we say that it has a finite presentation, if it is both finitely generated and finitely related. It is clear that every finite group has a finite presentation (although the con- verse is false). Simply take X = and, as relations, take all expressions of the form αiαj = αk for all αi, αj ∈ . In other words, the multiplication table of serves as the defining relations. It is well to point out that removing relations from a presentation of a group in general gives a larger group, the extreme case being that of the free group which has only generators and no relations. The simplest free group is the infinite cyclic group that has the presentation α with just one generator and no defining relation, whereas the cyclic group of order n has the presentation α|αn = 1 ; this group is denoted by Zn. The group with presentation α, β
- 29. 1.3 Graphs andgroups 7 is the infinite free group on two elements. The dihedral group of degree n is denoted by Dn. It has order 2n and also has a presentation with two generators: α, β|α2 = 1, βn = 1, α−1 βα = β−1 . Determining a group from a given presentation is not an easy problem. The reader who doubts this can try to show that the presentations α, β : αβ2 = β3 α, βα2 = α3 β and α, β, γ : α3 = β3 = γ 3 = 1, αγ = γ α−1 , αβα−1 = βγβ−1 both give the trivial group. We shall of course make a very simple use of stan- dard group presentations where these difficulties do not arise. The book [159] is a standard reference for advanced work on group presentations. The reader is referred to [147, 222] for any terms and concepts on group theory that are used but not defined in these chapters and, in particular, to [49, 62] for more information on permutation groups. 1.3 Graphs and groups Let G, G be two graphs. A bijection α : V(G) → V(G ) is called an isomor- phism if {u, v} ∈ E(G) ⇔ {α(u), α(v)} ∈ E(G ). The graphs G, G are, in this case, said to be isomorphic, and this is denoted by G G . Similarly, if D, D are digraphs, then a bijection α : V(D) → V(D ) is called an isomorphism if (u, v) ∈ A(D) ⇔ (α(u), α(v)) ∈ A(D ), and in this case the digraphs D, D are also said to be isomorphic, and again this is denoted by D D . If the two graphs, or digraphs, in this definition are the same, then α is said to be an automorphism of G or of D. The set of automorphisms of a graph or a digraph is a group under composition of functions, and it is denoted by Aut(G) or Aut(D). Note that an automorphism α of G is an element of SV(G), although it is its induced action on E(G) that determines whether α is an automorphism. This fact, although clear from the definition of automorphism, is worth emphasising when beginning to study automorphisms of graphs.
- 30. 8 Graphs andGroups: Preliminaries Figure 1.2. No automorphism permutes the edges as (12 23 34) For example, for the graph in Figure 1.2, the permutation of edges given by (12 23 34) is not induced by any permutation of the vertex-set {1, 2, 3, 4}. The only automorphisms for this graph are the identity and the permutation (14)(23), which induces the permutation (12 34)(23) of the edges in the graph. The question of edge permutations not induced by vertex permutations will be considered in some more detail later in this chapter. The process of obtaining a permutation group from a digraph can be reversed in a very natural manner. Suppose that (, Y) is a group of permutations acting on a set Y. Let A be a union of orbits of (, Y×Y). Clearly, the digraph D whose vertex-set is Y and whose arc-set is A has as a subgroup of its automorphism group. It might, however, happen that Aut(G) is larger than . Moreover, if the pairs in A are such that, for every (u, v) ∈ A, (v, u) is also in A, then replacing every opposite pair of arcs of D by a single edge gives a graph G such that ⊆ Aut(G). This and other ways of constructing graphs or digraphs admitting a given group of permutations will be studied in more detail in Chapter 4. Certain facts about automorphisms of graphs and digraphs are very easy to prove and are therefore left as exercises: (i) Aut(G) = Aut(G); (ii) Aut(G) = SV(G) if and only if G or G is Kn, the complete graph on n vertices; (iii) Aut(Cn) = Dn. Also, let α be an automorphism of G and u, v vertices of G. Then, (iv) deg(u) = deg(α(u)); (v) G − u G − α(u); (vi) d(u, v) = d(α(u), α(v)), where d(u, v) is the distance between u and v. Also, if u is a vertex in a digraph D and α is an automorphism of D, then (vii) degin(u) = degin(α(u)) and degout(u) = degout(α(u)). If u and v are vertices in a graph G and there is an automorphism α of G such that α(u) = v, then u and v are said to be similar. If G − u G − v, then u and v are said to be removal-similar. Property (v) tells us that if two vertices are similar, then they are removal-similar. The converse of this is, however,
- 31. 1.3 Graphs andgroups 9 false, as can be seen from the graph shown in Figure 1.3. Here, the vertices u, v are removal-similar but not similar. Such vertices are called pseudosimi- lar. Similar, removal-similar and pseudosimilar edges are analogously defined: Two edges ab, cd of G are similar if there is an automorphism α of G such that α(a)α(b) = cd. We shall be studying pseudosimilarity in more detail in Chapter 5. Sometimes we ask questions of this type: how many graphs (possibly of some fixed order n) are there? The answer to this question depends heavily on how we consider two graphs to be different. In general, if the order of a graph G is n, we can think of its vertices as being labelled with the integers {1, 2, . . . , n}. Two graphs G and H of order n so labelled are called identical or equal as labelled graphs (written G = H) if ij ∈ E(G) ⇔ ij ∈ E(H). (Compare this definition with that of isomorphic graphs.) Obviously, identical graphs are isomorphic, but the converse is not true. For example, the graphs in Figure 1.4 are isomorphic but not identical. Counting nonisomorphic graphs is, in general, much more difficult than counting nonidentical graphs. For example, there are four nonisomorphic graphs on three vertices but eight nonidentical ones. These are shown in Figures 1.5 and 1.6, respectively. Figure 1.3. A pair of pseudosimilar vertices Figure 1.4. Isomorphic but nonidentical graphs Figure 1.5. The four nonisomorphic graphs of order 3
- 32. 10 Graphs andGroups: Preliminaries Figure 1.6. The eight nonidentical graphs of order 3 Counting nonisomorphic graphs involves consideration of group symme- tries. For more on this the reader is referred to [103]. 1.4 Edge-automorphisms and line-graphs Although we shall be dealing mostly with Aut(G) and its realisation as the permutation group (Aut(G), V(G)), let us briefly look at other related groups associated with G. In this section we shall assume that G is a nontrivial graph, that is, its edge-set is nonempty. An edge-automorphism of a graph G is a bijection θ on E(G) such that two edges e, f are adjacent in G if and only if θ(e), θ(f) are also adjacent in G. The set of all edge-automorphisms of G is a group under composition of functions, and it is denoted by Aut1(G). The concept of edge-automorphisms can perhaps be best understood within the context of line-graphs. The line-graph L(G) of a graph G is defined as the graph whose vertex-set is E(G) and in which two vertices are adjacent if and only if the corresponding edges are adjacent in G. An automorphism of L(G) is clearly an edge-automorphism of G and (Aut1(G), E(G)) is equivalent to (Aut(L(G), V(L(G))). In this section we shall give the exact relationship between Aut1(G) and Aut(G), that is, between the automorphism groups of G and L(G). As we described earlier, any automorphism α of G naturally induces a bijec- tion α̂ on E(G) defined by α̂(uv) = α(u)α(v). It is an important (and easy to
- 33. 1.4 Edge-automorphisms andline-graphs 11 verify) property of α̂ that two edges e1, e2 are adjacent if and only if α̂(e1), α̂(e2) are adjacent, that is, if and only if α̂ is an edge-automorphism. For this reason α̂ is called an induced edge-automorphism of G. The set of all induced edge-automorphisms of G is denoted by Aut∗ (G), and it is easy to check that this is a subgroup of Aut1(G) under composition of functions. Now, it seems natural to expect that Aut(G) and Aut∗ (G) are iso- morphic. However, it can happen that two different automorphisms of G induce the same edge-automorphism. For example, let G = K2. Then |Aut(G)| = 2 but |Aut∗(G)| = 1. Also, suppose that G contains isolated vertices. Then any automorphism of G that permutes the isolated vertices and leaves all of the oth- ers fixed induces the trivial edge-automorphism. The following theorem says that these are basically the only situations when Aut(G) Aut∗ (G). Theorem 1.2 Let G be a nontrivial graph. Then Aut(G) Aut∗(G) if and only if G has at most one isolated vertex and K2 is not a component. Proof Clearly, the mapping α → α̂ is a homomorphism from Aut(G) onto Aut∗(G) because α̂.β̂(uv) = α.β(u)α.β(v) = αβ(uv). We must therefore show that the kernel of this mapping is trivial if and only if G has at most one isolated vertex and K2 is not a component. Suppose first that G has two isolated vertices u, v or K2 as a component with vertices u, v. Then the permutation α that transposes u and v and fixes all of the other vertices is a nontrivial automorphism of G, but α̂ is the identity. Therefore the kernel is not trivial. Conversely, suppose that G does not contain K2 as a component nor its com- plement. If Aut(G) is trivial, then so is Aut∗(G). Therefore, let α be a nontrivial element of Aut(G), and let α(u) = v = u. Then deg(u) = deg(v) = 0 (other- wise u, v would be a pair of isolated vertices). We consider two cases. Case 1: u, v adjacent. Let e be the edge uv. Then deg(u) = deg(v) 1 (otherwise the two vertices u, v would form a component K2). Therefore, there exists an edge f = e incident to u (but not to v, since the graph is simple). But α̂(f) must be incident to v (since α(u) = v), that is, α̂(f) = f, and hence α̂ is not trivial. Case 2: u, v not adjacent. Let e be an edge incident to u. Again, e is not incident to v but α̂(e) is. Therefore α̂ is again nontrivial. The next natural question to ask is whether there can be edge-automorphisms of G that are not induced by automorphisms, that is, whether Aut∗(G) can be a strict subgroup of Aut1(G). This situation can very well happen, although, as we shall see, such cases are quite rare.
- 34. 12 Graphs andGroups: Preliminaries Figure 1.7. Graphs with edge-isomorphisms not induced by isomorphisms Before proceeding let us first extend the idea of edge-automorphisms on the edge-set of a graph to that of edge-isomorphisms between edge-sets of different graphs. Let G, G be two nontrivial graphs. A bijection θ : E(G) → E(G ) is an edge-isomorphism if e, f adjacent in G ⇔ θ(e), θ(f) adjacent in G . Two graphs are said to be edge-isomorphic if there is an edge-isomorphism between their edge-sets. The graphs W1, W2 in Figure 1.7 are edge-isomorphic, although they are not isomorphic. That is, there is an edge-isomorphism between their edge- sets that cannot be induced by an isomorphism between their vertex-sets. This means that their line-graphs are isomorphic even though the two graphs are not themselves isomorphic. Also, each of the graphs W3, W4, W5 in the same figure has edge- automorphisms that are not induced by automorphisms. That is, the group Aut∗ (Wi) is a strict subgroup of Aut1(Wi). In other words, Aut(L(Wi)) is larger than Aut(Wi). The following theorem of Whitney [258] says that these are essentially the only cases when edge-isomorphisms that are not induced by isomorphisms can arise. We give the statement of the theorem without proof, which, although not deep or difficult, would lengthen this introductory chapter without adding significant new insights. Theorem 1.3 (Whitney) Let G, G be connected graphs different from the five graphs in Figure 1.7. Let θ : E(G) → E(G ) be an edge-isomorphism. Then θ is induced by an isomorphism from G to G .
- 35. 1.5 A wordon issues of computational complexity 13 Whitney’s Theorem and Theorem 1.2 together therefore give the following corollary. Corollary 1.4 Let G be a nontrivial graph. Then Aut1(G) = Aut∗(G) if and only if both of these conditions hold: (i) not both W1, W2 are components of G; (ii) none of Wi, i = 3, 4, 5 is components of G. Moreover, Aut1(G) Aut(G) (that is, Aut(L(G)) Aut(G)) if and only if (i) and (ii) hold and G has at most one isolated vertex and K2 is not a component of G. 1.5 A word on issues of computational complexity Although in this book we shall not concern ourselves with issues of computa- tional complexity, it is perhaps worthwhile to say a few words in this regard here in order to put matters into a better perspective. A student reading the def- initions of isomorphic graphs and automorphisms might think that it is an easy matter to determine in general whether two given graphs are isomorphic or to compute the automorphism group of a graph. In fact, this is far from being the case, and these problems are very hard to crack in practice, at least as far as present knowledge goes. In general, one considers that an efficient algorithm exists for finding a solu- tion to a problem (for example, finding a nontrivial automorphism of a given graph) if there is a general algorithm such that the number of operations that it takes to solve the problem is a polynomial function of the size of the input (say, the number of vertices in the graph); one says that the algorithm solves the problem in polynomial time. Of course, several terms in the previous sentence need exact definitions, but we shall here take an intuitive approach and refer the reader to [36] or [82] for the exact details on computational complexity. Those problems for which an efficient (polynomial-time) algorithm exists form the class denoted by P (which stands for ‘polynomial’). However, there are several problems for which it is not known whether an efficient algorithm does exist. In order to tackle this question of computational intractibility, two important ideas have been developed. Firstly, the class NP (which stands for ‘nondeterministic polynomial’) is defined. Roughly (again we refer the reader to the textbooks cited earlier for the exact details) this class contains all of those problems for which, given a candidate solution, one can verify in polynomial time that it is in fact a correct
- 36. 14 Graphs andGroups: Preliminaries solution. For example, the problem of determining whether a graph has a non- trivial automorphism is in NP, since, given such a permutation of the vertices, it is easy to determine in polynomial time that it is an automorphism. Now the main question in computational complexity is whether P = NP (clearly P ⊆ NP), and to tackle this question another important idea is intro- duced. Given two problems A and B, one says that A is (polynomially) reducible to B if, given an algorithm for solving B, it can be transformed in polynomial time into an algorithm for solving A. Reducibility therefore introduces a hier- archy between problems for, if A is reducible to B, then, in a sense, A cannot be more difficult to solve (computationally) than B. In particular, if there is an efficient algorithm for solving B, then there is also an efficient algorithm for solving A. Now, the question of reducibility took on special significance by the discov- ery that in the class NP there are problems, called NP-complete, to which any other problem in NP is reducible. In other words, if an efficient algorithm can be found for any NP-complete problem, then all problems in NP would have an efficient algorithm to solve them, and P would be equal to NP. Now, it is not known whether the problem of determining if two graphs are isomorphic, which lies clearly in NP, is NP-complete. In fact, if it turns out that P is not equal to NP, then there is evidence to suggest that the problem of graph isomorphism might lie strictly between the classes P and NP. What all this means in practice is that, as far as present knowledge goes, no general algorithm can determine in a guaranteed reasonable time whether two graphs are isomorphic, or whether a given graph has a nontrivial automorphism (these two problems are closely related [126]). It is known that for special types of graphs (for example, trees, planar graphs and graphs with bounded degree) an efficient algorithm does exist. Computer packages can also help one to solve these problems, certainly more efficiently than an attempt ‘by hand’ for large graphs, although their time performance is not guaranteed (by what we said earlier). For example, the soft- ware package MathematicaTM has a combinatorics extension that, amongst other things, finds graph automorphisms and isomorphisms. A more specialised package, and one that is freely available from www.combinatorialmath.org.ca/gg/index.html is Groups Graphs [131], developed by Bill Kocay. This package contains several combinatorial routines related to graphs, digraphs, combinatorial designs and their automorphism groups and also embeddings of graphs on some sur- faces and a graph isomorphism algorithm. It is easy to use, and it has a pleasant graphical user interface. It is also very useful simply for drawing diagrams of
- 37. 1.6 Exercises 15 graphs.Although originally written for MacintoshTM computers, a version for the unix-based Haiku operating system is in preparation, and this version will contain several new features. An important computer algebra package, which is also freely available, is the system GAP [243]. This package performs very sophisticated routines in discrete abstract algebra, in particular routines on permutation groups. It incor- porates a number of extensions, one of which, GRAPE [235], deals specifically with graphs, including their automorphisms and isomorphisms. The computer package Sage [227] is an open-source competitor to systems like MapleTM, MathematicaTM and MatlabTM. It incorporates several open- source mathematical software like GAP and R, and it can be run via Sage- MathCloud without the need of installing the system on one’s computer. It has an excellent library of functions for doing graph theory. In this book we shall present some constructions using GAP and Sage. Finally, it should be mentioned that it is generally accepted that the best package to tackle graph isomorphisms is nauty [181], developed by Brendan McKay. In fact, the system GRAPE invokes nauty when computing automor- phisms or isomorphisms. 1.6 Exercises 1.1 Draw all twenty nonidentical graphs with vertex-set {1, 2, 3, 4} that have three edges. How many of them are nonisomorphic? In general, how many nonidentical graphs on n vertices and m edges are there? How many are there on n vertices? 1.2 Let G be the graph in Figure 1.8. How many nonidentical labellings does G have using the labels {1, 2, . . . , 6} on its vertices? In general, how many nonidentical labellings does a graph G on n vertices have using the labels {1, 2, . . . , n} on its vertices? 1.3 Show that if G is self-complementary (that is, G G), then n ≡ 0 mod 4 or n ≡ 1 mod 4. Determine all self-complementary graphs on five vertices. 1.4 A well-known result due to Cayley says that the number of nonidentical trees on n vertices is nn−2. Verify this for n = 4. Look up one of the several proofs of this result. Figure 1.8. How many distinct labellings does this graph have?
- 38. 16 Graphs andGroups: Preliminaries The points 1, 2, . . . , n are drawn in a plane. A random tree is drawn joining these points, with all possible spanning trees being equally likely. Let pn be the probability that 1 is an endvertex of the tree. Show that limn→∞ pn = 1/e. 1.5 Find a graph with a pair of pseudosimilar edges. 1.6 Let Y = {1, 2, 3, 4} and let be the group acting on Y generated by the permu- tation (1 2 3 4). Construct a digraph D whose vertex-set is Y and whose arc-set is the orbit of the arc (1, 2) under the action of the permutation group (, Y × Y). Is the whole of Aut(D)? Can a graph be obtained by taking the orbit of some other arc or a union of orbits? Will always be the whole of Aut(D)? 1.7 Let P be a rectangular plate and Q a plate in the form of a rhombus. Show that the groups of symmetry of P and Q are isomorphic as abstract groups but not equivalent considered as permutation groups of the four vertices of P and Q. Show that the same situation arises with the following two graphs: the cycle on four vertices with an extra multiple edge and the complete graph K4 with an edge deleted. 1.8 Show that the dihedral group Dn can be presented as α, β|α2 = β2 = (αβ)n = 1 . 1.9 Let 1 be the abelian group defined by the presentation α, β, γ |α3 = β3 = γ 3 = 1, [α, β] = [α, γ ] = [β, γ ] = 1 , where [α, β] = α−1β−1αβ is the commutator of α and β, and let 2 be the group defined by the presentation α, β, γ |α3 = β3 = γ 3 = 1, [β, α] = γ , [α, γ ] = [β, γ ] = 1 . Show that although 1 and 2 are not isomorphic as abstract groups, and there- fore the two permutation groups (L(1), 1) and (L(2), 2) are not equivalent, still the cycle structures of the permutations of L(1) acting on 1 are the same as those of the permutations of L(2) acting on 2. Give an example of two permutation groups whose permutations have the same cycle structures and which are isomorphic as abstract groups but are still not equivalent as permutation groups. 1.10 Prove Theorem 1.1. 1.11 This exercise is intended to illustrate Theorem 1.1. Consider the action of the alternating group A4 on the set X = {1, 2, 3, 4}. Let H be the stabiliser of the element 1 under this action. Confirm that the left action of A4 on the left cosets of H is equivalent to the action of A4 on X. 1.12 Let be a group of permutations acting on a set Y and let y, x be two elements of Y that are in the same orbit under this action. Let α, β ∈ be two permutations such that α(y) = β(y) = x. Prove that αy = βy. Prove also that if x is in the same orbit as y and γ is a permutation such that γ (y) = x and γ y = αy, then x = x. 1.13 Show that if the abelian permutation group acts transitively on Y, then its action on Y is regular. 1.14 (a) Show that a 2-transitive permutation group is primitive. (b) Show that if Aut(G), for a graph G, is 2-transitive, then either G or its com- plement is a complete graph.
- 39. 1.7 Notes andguide to references 17 (c) Suppose that the transitive permutation group (, Y), with |Y| finite, is imprim- itive. Show that the blocks of an imprimitivity block system have equal size. 1.15 Let G be a vertex-transitive graph whose automorphism group acts impri- mitively on V(G). Show that the subgraphs of G induced by the blocks of an imprimitivity block system are all isomorphic. Suppose that each such subgraph is replaced by its complement, leaving the other edges intact. Let G be the resulting graph. Show that Aut(G ) = Aut(G). 1.16 The Petersen graph can be defined as follows. Let N = {1, 2, 3, 4, 5}, and let the vertices of the graph be all subsets of N of size 2 in which two vertices are adja- cent if the corresponding subsets are disjoint. Use this definition and GAP (with GRAPE) to construct the Petersen graph and to verify some of its properties. 1.7 Notes and guide to references One of the standard texts on graph theory has, for many years, been [97]. More recent books that give an excellent coverage of the subject are [28, 61, 257, 259]. The last reference is a short introduction that is quite sufficient back- ground for this book. Biggs’ book [24] is the standard text on algebraic graph theory, but the more recent [90] is also an excellent and up-to-date textbook on the subject. The book [94] contains a number of recent and specialised survey papers on various aspects of algebraic graph theory, particularly those dealing with graph symmetries. A proof of Whitney’s Theorem can be found in [22]. We shall need only the most elementary notions of group theory. The text [147] gives ample coverage for our purposes, while [222] provides a more complete treatment. Two excellent books devoted entirely to permutation groups are [49, 62]. Most of the results and definitions on permutation groups that we have given here and others that we shall need can be found in the first few chapters of these two books. For a full discussion of the terms on computational complexity that were introduced earlier rather intuitively, the reader is referred to the standard text- book [82] or the more recent [36]. The book [126] and the references it cites are suggested for those who are interested in the computational complexity of the graph isomorphism problem. Those who are particularly interested in some of the powerful algebraic techniques used to tackle this problem should look at the papers [106, 155]. For practical computations on a computer with per- mutation groups and graph automorphisms and isomorphisms in particular, the systems [131, 181, 235, 243] are recommended.
- 40. 2 Various Types ofGraph Symmetry We shall see in this chapter that most graphs are asymmetric, that is, their automorphism group is trivial; in other words, it consists only of the identity permutation. The least number of vertices that an asymmetric graph can have is six, and the graph shown in Figure 2.1 is the smallest such graph in the sense that any other asymmetric graph on six vertices has more edges. As is to be expected, however, the most interesting relationships between groups and graphs arise when the graphs have a very high degree of symmetry, that is, a large automorphism group. One way to make more precise the idea of a large automorphism group is to require that it at least be transitive on the vertex-set or the edge-set of the graph. The weakest forms of symmetry to ask of a graph involve vertex-transitivity and edge-transitivity, which we define in the next section. 2.1 Transitivity Recall the definition of similar vertices from the previous chapter. We say that a graph G is vertex-transitive if any two vertices of G are similar, that is, if, for any u, v ∈ V(G), there is an automorphism α of G such that α(u) = v. In other words, G is vertex-transitive if all of the vertices of G are in the same orbit of the permutation group (Aut(G), V(G)). Figure 2.1. The smallest asymmetric graph 18
- 41. 2.1 Transitivity 19 Figure2.2. A vertex-transitive graph that is not edge-transitive One can define edge-transitivity analogously. A graph G is edge-transitive if, given any two edges {a, b} and {c, d}, there exists an automorphism α such that α{a, b} = {c, d}, that is, {α(a), α(b)} = {c, d}. In other words, G is edge- transitive if any two of its edges are similar under the action of the permutation group (Aut∗(G), E(G)), that is, if the edges of G are all in one orbit under this action. Note that very often the word ‘transitive’ is used to refer to a graph, and in this case it is taken to mean ‘vertex-transitive’. Vertex-transitivity does not imply edge-transitivity, nor does the converse implication hold. Figure 2.2 shows a graph that is vertex-transitive but not edge-transitive. The complete bipartite graph Kp,q with p = q is a simple example of a graph that is edge-transitive but not vertex-transitive. The following well-known result does give a description of edge-transitive graphs that are not vertex- transitive. Theorem 2.1 Let G be a graph without isolated vertices and let H be a sub- group of Aut(G). Suppose that the action induced by H is transitive on the edges of G but not on its vertices. Then G is bipartite and the action of H on V(G) has two orbits that form the bipartition of V(G). Proof Let {u, v} be an edge of G. Let V1, V2 be the orbits under the action of H containing u and v, respectively. (We are not excluding, for the moment, the possibility that V1 = V2.) Let x be any other vertex of G. Since G does not have isolated vertices, there exists a vertex y adjacent to x; that is, {x, y} is an edge of G. But G is edge-transitive under the action of H; therefore the two edges are similar under this action. Hence x is similar to at least one of u or v, that is, x is in V1 or V2. Therefore V1 ∪ V2 = V(G). Now, V1, V2 must be disjoint; otherwise (since orbits form a partition) they are equal, and this would mean that the vertices of G are all in one orbit, giving that G is vertex-transitive under the action of H. Hence we now have that the action of H on V(G) has exactly two orbits, V1, V2.
- 42. 20 Various Typesof Graph Symmetry Now let a, b be in the same orbit, say V1. It then follows that a, b are not adjacent. For suppose otherwise. Then the edge {u, v} is similar to the edge {a, b} under the action of H; therefore the vertex v is similar to one of a or b under this action, giving that v (which is in the orbit V2) is also in the orbit V1, which contains a and b. But this is impossible since V1 ∩ V2 = ∅. Hence, as required, we have that no two vertices from the same orbit can be adjacent. Corollary 2.2 If a graph G without isolated vertices is edge-transitive but not vertex-transitive, then it is bipartite and the action of Aut(G) on V(G) has two orbits that form the bipartition of V(G). Proof Take H = Aut(G) in the previous theorem. 2.1.1 Semisymmetric graphs The typical example of edge-transitive but not vertex-transitive graphs given earlier is the complete bipartite graph with a different number of vertices in the bipartition. These graphs are trivially not vertex-transitive because their vertices have different degrees. Although regular graphs do exist that are edge- transitive but not vertex-transitive, it is quite difficult to construct them. Such graphs are now called semisymmetric graphs and they were first studied by Folkman [77], who constructed the smallest possible semisymmetric graph having twenty vertices.1 One construction of the Folkman Graph is described in Exercise 2.5. Here we shall describe another well-known semisymmetric graph, the Gray Graph.2 For a long time nobody could find a smaller cubic semisymmetric graph, and eventually it was formally proved in [160] that it is the smallest cubic semisymmetric graph. We shall see a more systematic way of describing it in subsequent chapters. Here we follow Bouwer’s construction in [33]. Consider a cycle on 54 vertices numbered consecutively from 0 to 53. To form the Gray Graph G add the following edges to this cycle: {1, 42}, {2, 15}, {3, 28}, {4, 33}, {5, 44}, {6, 53}, {7, 48}, {8, 21}, {9, 32}, {10, 45}, {11, 24}, {12, 41}, {13, 20}, {14, 31}, {16, 35}, {17, 40}, {18, 49}, {19, 0}, {22, 51}, {23, 30}, {25, 38}, {26, 43}, {27, 52}, {29, 36}, {34, 47}, 1 More information about this graph, called the Folkman Graph, can be found on the MathWorld page http://mathworld. wolfram.com/FolkmanGraph.html 2 The Gray Graph is also featured on MathWorld at http://mathworld.wolfram.com/Gray Graph.html
- 43. 2.1 Transitivity 21 {37,50}, {39, 46}. It is tedious but not difficult to check that the permutations α = (2 0 43)(3 53 43)(4 6 44)(7 45 33)(8 10 32)(11 31 21) = (12 14 20)(15 19 41)(16 18 40)(22 24 30)(25 29 51) = (26 28 52)(34 48 46)(35 49 39)(36 50 38) and β = (1 7 11 37 15 53 9 25 35)(2 6 10 38 16 0 8 24 36) = (3 5 45 39 17 19 24 23 29)(4 44 46 40 18 20 22 30 28) = (12 50 14 52 32 26 34 42 48)(13 51 31 27 33 43 47 41 49) are automorphisms of G. Note that the automorphism α fixes the vertex 1 and permutes cyclically its neighbours 2, 42 and 0. Thus, in order to show that the graph is edge-transitive it is sufficient to show that any odd-numbered vertex can be mapped into 1 by an automorphism of G. This can be done by appropriate products of α and β. For example, α4β maps vertex 53 to vertex 1. However, the graph is not vertex-transitive because from an odd-numbered vertex it is possible to have three different paths of length 4 joining the vertex to some other common vertex (for example, vertex 1 to vertex 5), but this is not possible from an even-numbered vertex. Another way to show that the Gray Graph is not vertex-transitive is to con- sider the distance sequences of its vertices [166]. The distance sequence of a vertex v is the vector (a0, a1, . . . , ar) where ai is the number of vertices at dis- tance i from v. In the case of the Gray Graph, the distance sequences of the vertices in the two colour classes are (1, 3, 6, 12, 12, 12, 8) and (1, 3, 6, 12, 16, 12, 4), respectively, therefore these vertices cannot be in the same orbit under the automorphism group of the graph. Although the Sage package has the Gray Graph already implemented, it is easy and instructive to show how to construct it following the specification given earlier. First one creates the vertex-set which will be the list of numbers from 1 to 54. Sage, like many computer languages such as Python, starts its lists from 0. Therefore a list of length n produced by the command range(55) would contain the numbers from 0 to 54. To start the list from 1 we have to define the list of vertices as vertices := range(1,55);
- 44. 22 Various Typesof Graph Symmetry The graph is then constructed first using the command DiGraph so that we do not need to repeat every pair of adjacent vertices twice. This com- mands basically takes two parameters. The first parameter is the vertex-set of the graph to be constructed, and the second parameter is a boolean function (defined with the lambda construct) of two variables which compares all pos- sible pairs of the vertex-set and an edge is drawn between any pair of vertices for which the function returns True. dgray := DiGraph([vertices, lambda i, j: (i == mod[j + 1, 54]) or (i == 53 and j == 54) or (i == 1 and j == 42) or (i == 2 and j == 15) or (i == 3 and j == 28) or (i == 4 and j == 33) or (i == 5 and j == 44) or (i == 6 and j == 53) or (i == 7 and j == 48) or (i == 8 and j == 21) or (i == 9 and j == 32) or (i == 10 and j == 45) or (i == 11 and j == 24) or (i == 12 and j == 41) or (i == 13 and j == 20) or (i == 14 and j == 31) or (i == 16 and j == 35) or (i == 17 and j == 40) or (i == 18 and j == 49) or (i == 19 and j == 54 or (i == 22 and j == 51) or (i == 23 and j == 30) or (i == 25 and j == 38) or (i == 26 and j == 43) or (i == 27 and j == 52) or (i == 29 and j == 36) or (i == 34 and j == 47) or (i == 37 and j == 50) or (i == 39 and j == 46) ] ) This digraph is then changed into an undirected graph with the following command which changes every arc into an edge.
- 45. 2.1 Transitivity 23 gray= dgray.to_undirected() It is then easy to check that the aforementioned properties of the Gray Graph hold. For example, in order to check whether it is vertex-transitive we use the command gray.is_vertex_transitive() which returns False. The command gray.is_edge_transitive() returns True, as expected, while the command gray.is_regular() also returns True, confirming that the graph is semisymmetric. In fact, we could have reached the same conclusion with the command gray.is_semi_symmetric() which also returns True. Finally, one can check whether the graph constructed earlier is isomorphic to Sage’s inbuilt ‘GrayGraph’ using the command graphs.GrayGraph.is_isomorphic(gray) which, again, returns True. We shall have more to say about the Gray Graph in a later chapter when we shall describe it in a more algebraic fashion. Exercises 2.2 and 2.5 show that any semisymmetric graph must have even order and its degree must be less than |V(G)|/2. 2.1.2 Arc-transitive and 1 2 -arc-transitive graphs A stronger form of transitivity than either vertex- or edge-transitivity based on the edge-set of G can also be defined. If G has the property that, for any two edges {a, b}, {c, d}, there is an automorphism α such that α(a) = c and α(b) = d and also an automorphism β such that β(a) = d and β(b) = c, then G is said to be arc-transitive. We shall now derive a result of Tutte that gives a restriction on the degree of the vertices of a graph that is vertex-transitive and edge-transitive but not arc-transitive. One can think of arc-transitivity as follows. Given any graph G, construct the directed graph ← → G obtained from G by replacing each edge {a, b} by the pair of
- 46. 24 Various Typesof Graph Symmetry arcs (a, b) and (b, a). Then clearly Aut(G) = Aut( ← → G ) and any automorphism α of G induces the natural action on arcs given by (a, b) → (α(a), α(b)). Then G is arc-transitive precisely if, given any two arcs in ← → G , there is an automorphism of ← → G mapping one arc into the other. This is a stronger form of transitivity than both vertex- and edge-transitivity because now, given any two edges on each of which an orientation is imposed, there is an automorphism mapping one edge into the other and preserving the given orientations. In fact, an arc-transitive graph is both vertex-transitive and edge-transitive. Lemma 2.3 Let H be a subgroup of Aut(G) such that, under the action of H, G is vertex-transitive and edge-transitive but not arc-transitive. Let t be an arc of ← → G and let D be the subdigraph of ← → G whose vertex-set is V( ← → G ) and whose arc-set is the orbit of t under the action of H. Then (i) for every edge {a, b} of G, D contains exactly one of the arcs (a, b) or (b, a); (ii) H ≤ Aut(D); (iii) D is vertex-transitive. Proof (i) Let t = (s1, s2). Because G is edge-transitive under the action of H, there is an α ∈ H such that α{s1, s2} = {a, b}. Therefore certainly one of (a, b) or (b, a) is an arc of D. Suppose that both are arcs of D. Then there is some β ∈ H such that β((a, b)) = (b, a). But given any edge {c, d} of G there is, by edge-transitivity, a γ ∈ H such that γ ((a, b)) equals (c, d) or (d, c). Suppose, without loss of generality, that γ ((a, b)) = (c, d). But then γβ((a, b)) = (d, c). Therefore, for any edge {c, d} of G, both arcs (c, d) and (d, c) are in the same orbit, that is, the action of H on G is arc-transitive, a contradiction. (ii) This follows because the arc-set of D is a full orbit of the permutation group (H, V(G) × V(G)). (iii) This follows because (H, V(G)) is transitive, V(D) = V(G) and H ≤ Aut(D). If we let H = Aut(G) in this lemma, then, in view of (i), if G is a vertex- transitive and edge-transitive graph that is not arc-transitive, it follows that the arc-set of ← → G is naturally partitioned into two orbits of equal size under the action of Aut(G), and none of the two orbits contains both an arc (a, b) and its inverse (b, a). In view of this, a graph that is vertex-transitive, edge-transitive but not arc-transitive is said to be 1 2 -arc-transitive.
- 47. 2.2 Asymmetric graphs25 Figure 2.3. Relationship between different types of transitivity The relationship between these forms of transitivity is shown in Figure 2.3, where a line leading down from one property to another means that the first implies the second. Although 1 2 -arc-transitive graphs are not easy to find, they do exist (an exam- ple will be given in Chapter 3). The following well-known theorem of Tutte tells us that such a graph must have even degree. Theorem 2.4 (Tutte) Let H be a subgroup of Aut(G) such that, under the action of H, G is vertex-transitive and edge-transitive but not arc-transitive. Then the degree of G is even. In particular, a 1 2 -arc-transitive graph has even degree. Proof Let D be as in the previous lemma. By the third part of this lemma, all vertices of D have the same out-degree, say k. Now, k·|V(D)| = |A(D)| and, by the first part of the lemma, |A(D)| = |E(G)|. But if the common degree of the vertices of G is d, then, by the Handshaking Lemma, |E(G)| = d · |V(G)|/2 = d · |V(D)|/2. Therefore d = 2k, that is, d is even. 2.2 Asymmetric graphs Although we shall be mostly interested in graphs with nontrivial automorphism groups, let us briefly consider asymmetric graphs. Let P be a graph theoretic property such as ‘planar’ or ‘vertex-transitive’. Let rn denote the proportion of
- 48. 26 Various Typesof Graph Symmetry labelled graphs on n vertices that have property P. If limn→∞ rn = 1, then we say that almost every (a.e.) graph has property P. We have already said that almost every graph is asymmetric. We shall soon prove a stronger result that will be used in a later chapter when we consider the Reconstruction Problem. The following probability space is often set up when studying random graphs. Let G(n, p) be the set of all labelled graphs on the set of vertices {1, 2, . . . , n} where, for each pair i, j, P(ij is an edge) = p and P(ij is not an edge) = 1 − p independently. Therefore a graph with m edges in G(n, p) has probability pm q(n 2)−m , where q = 1 − p. We shall need only this space when the probability p = 1 2 . In this case, each graph G in G(n, 1 2 ) has probability (1 2 )(n 2), which is, of course, equal to the probability of choosing G randomly from amongst all 2(n 2) labelled graphs on n vertices when all are equally likely to be chosen. Therefore, to show that a.e. graph has a particular property P one has to show that the probability that G ∈ G(n, 1 2 ) has property P tends to 1 as n tends to infinity. Now, let k be fixed. We say that a graph G has property Ak if all induced subgraphs of G on n − k vertices are mutually nonisomorphic. In other words, G has property Ak means that, if X, Y are two distinct k-subsets of V(G), then G − X G − Y. It is easy to show (Exercise 2.5) that if G has property Ak+1, then it also has property Ak and that if it has property A1, then it is asymmetric. We shall show that, for any fixed k, a.e. graph has property Ak. Lemma 2.5 Let W ⊆ V, |W| = t, |V| = n, and let ρ : W → V be an injective function that is not the identity. Let g = g(ρ) be the number of elements w ∈ W such that ρ(w) = w. Then there is a set Iρ of pairs of (distinct) elements of W, containing at least 2g(t − 2)/6 pairs, such that Iρ ∩ ρ(Iρ) = ∅. Proof Consider those pairs v, w ∈ W such that at least one is moved. (All pairs are taken to contain distinct elements.) There are g(t − g) + g 2 such pairs. For all but at most g/2 of these pairs, {v, w} = {ρ(v), ρ(w)} (the exceptions are when ρ(v) = w and ρ(w) = v). Let Eρ be the set of all such pairs. Then |Eρ| ≥ g(t − g) + g 2 − g/2 = g(t − g/2 − 1) ≥ g(t/2 − 1).
- 49. 2.2 Asymmetric graphs27 Define a graph Hρ with vertex-set the pairs in Eρ and such that each pair {v, w} is adjacent to the pair {ρ(v), ρ(w)}. In Hρ, all degrees are at most 2. Degrees equal to 1 could arise because {ρ(v), ρ(w)} could contain an element not in W, and so the pair would not be in Eρ. Degrees equal to 2 could arise because {v, w} could be adjacent to both {ρ(v), ρ(w)} and {ρ−1(v), ρ−1(w)}. Therefore the components of Hρ are isolated vertices, paths or cycles. Let Iρ be a set of independent (that is, mutually not adjacent) vertices in Hρ. There- fore, for any pair {v, w} ∈ Iρ, {ρ(v), ρ(w)} is not in Iρ. Now, all isolated vertices in Hρ are independent, at least half of the vertices on a path are independent and at least one third of the vertices on a cycle are independent, the extreme case here being a triangle. Therefore |Iρ| ≥ |Eρ|/3 ≥ 2g(t − 2)/6, as required. Corollary 2.6 Let G ∈ G(n, 1 2 ), W ⊂ V = V(G) and |W| = t. Let ρ : W → V be an injective function that is not the identity. Let g = g(ρ) be the number of elements w ∈ W such that ρ(w) = w. Let Sρ be the event ‘ρ gives an isomorphism from G[W] to G[ρ(W)]’. Then P(Sρ) ≤ 1 2 2g(t−2)/6 . Proof Let Iρ be the set constructed in the previous lemma. Now, for a given pair {v, w} ∈ Iρ, the event ‘{v, w} and {ρ(v), ρ(w)} are both edges or nonedges’ has probability 1/2. These events, as they range over all pairs {v, w} ∈ Iρ, are mutually independent because they involve distinct pairs. But Sρ requires all these events simultaneously. Therefore, by independence, P(Sρ) ≤ 1 2 |Iρ| ≤ 1 2 2g(t−2)/6 , as required. The result of this corollary is the crux of the proof of the following theo- rem: There are too many independent correct ‘hits’ required for ρ to be an isomorphism, and the probability therefore becomes small as n increases.
- 50. 28 Various Typesof Graph Symmetry Theorem 2.7 (Korshunov; Müller; Bollobás) Let k be a fixed nonnegative integer and let G ∈ G(n, 1 2 ). Let pn denote the probability that ∃W ⊆ V(G) = V = {1, 2, . . . , n}, with |W| = n − k and such that ∃ρ : W → V, ρ = id, ρ is an isomorphism from G[W] to G[ρ(W)]. Then, limn→∞ pn = 0. Hence, a.e. graph has property Ak. Proof Pick a particular W ⊂ V with |W| = n − k. This can be done in n n−k ways, and n n − k = n(n − 1) . . . (n − k + 1) k! nk . Let t = n − k. Let ρ : W → V be injective and not the identity, and let g = g(ρ) be the number of vertices of W that are moved by ρ. Let Sρ be the event defined in the previous corollary. Now, for a given value of g between 1 and t, how many functions ρ are there such that g(ρ) = g? Such a function is determined by the set {w : ρ(w) = w} and by the values it takes on this set. Therefore, there are less than n2g such ρ. Therefore, for a given fixed W, the probability of a nontrivial isomorphism is given by ρ=id P(Sρ) = t g=1 ρ:g(ρ)=g P(Sρ) ≤ t g=1 n2g 1 2 2g(t−2)/6 = t g=1 n2 2(2−t)/3 g t g=1 41/3 n2 2−t/3 g .
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- 52. If relating, i.e.,cognition, is established on basis of inter-relation in brain tissue, if every mental connecting means a connecting of brain fibres, we might, indeed, determine the number of thoughts, but we could not tell what the thoughts were. So if mental disturbance always means bodily disturbance, we can still tell nothing more about the nature of each emotion than we knew before. We must first know fear, anger, etc., as experiences in consciousness before we can correlate them with corporeal acts. Is now this necessarily subjective method peculiarly limited as to feeling? Can we know feeling directly as psychic act or only indirectly through accompaniments? Mr. James Ward (vide article on Psychology in the Encyclopædia Britannica, p. 49, cf. p. 71) remarks that feelings cannot be known as objects of direct reflection, we can only know of them by their effects on the chain of presentation. The reason for this is, that feeling is not presentation, and “what is not presented cannot be re-presented.” “How can that which was not originally a cognition become such by being reproduced?” It cannot. But do we need to identify the known with knowing, in order that it may be known? Must feeling be made into a cognition to be cognized? It is obvious enough that no feeling can be revived into a representation of itself, but no more can any cognition or any mental activity. Revival or recurrence of consciousness can never constitute consciousness of consciousness which is an order apart. If cognition is only presentation and re-presentation of objects, we can never attain any apprehension of consciousness, any cognition of a cognition or of a feeling or of a volition, for they are all equally in this sense subjective acts. Re-presentation at any degree is never by itself sense of re-presentation or knowledge of the presentation. Of course, the doctrine of relativity applies to introspection as to all cognition, and subject qua subject is as unknowable as object qua object. We do not know feeling in itself, nor anything else in itself, the subjective like the objective ding an sich is beyond our ken. Yet kinds of consciousness are as directly apprehended and discriminated as kinds of things, but the knowing is, as such, distinct from the known even when knowing is known. Here the act knowing is not the act known and is different in value. The object known is
- 53. not, at leastfrom the purely psychological point of view, ever to be confounded with the knowing, to be incorporated into cognition by virtue of being cognized. Feeling, then, seems to be as directly known by introspection and reflection as any other process. It is not a hypothetical cause brought in by the intellect to explain certain mental phenomena, but it is as distinctly and directly apprehended as cognition or volition. The distinction between having a feeling and knowing a feeling is a very real one, though common phraseology confuses them. We say of a brave man, he never knew fear; by which we mean he never feared, never experienced fear, and not that he was ignorant of fear. Again, in like manner, we say sometimes of a very healthy person, he never knew what pain was, meaning he never felt pain. These expressions convey a truth in that they emphasize that necessity of experience in the exercise of the subjective method upon which we have already commented, but still they obscure a distinction which must be apparent to scientific analysis. We cannot know feeling except through realization, yet the knowing is not the realization. Being aware of the pain and the feeling pain are distinct acts of consciousness. All feeling, pain and pleasure, is direct consciousness, but knowledge of it is reflex, is consciousness of consciousness. The cognition of the pain as an object, a fact of consciousness, is surely a distinct act from the pain in consciousness, from the fact itself. The pain disturbance is one thing and the introspective act by which it is cognized quite another. These two acts are not always associated, though they are commonly regarded as inseparable. It is a common postulate that if you have a pain you will know it, or notice it. If we feel pained, we always know it. This seemingly true statement comes of a confounding of terms. If I have a pain, I must, indeed, be aware of it, know it, in the sense that it must be in consciousness; but this makes, aware of pain, and knowing pain, such very general phrases as to equal experience of pain or having pain. But there is no knowledge in pain itself, nor pain in the knowing act per se. The knowing the pain must be different from the pain itself, and is not always a necessary sequent. We may experience pain without
- 54. cognizing it assuch. When drowsy in bed I may feel pain of my foot being “asleep,” but not know it as a mental fact. We may believe, indeed, that pain often rises and subsides in consciousness without our being cognizant of it, but, of course, in the nature of the case there is no direct proof, for proof implies cognizance of fact. Pain as mental fact, an object for consciousness, not an experience in consciousness, is what is properly meant by knowing pain. Consciousness-of-pain as knowledge of it is not always involved by pain-in-consciousness as experience of it. Consciousness of pain by its double meaning as cognizance of pain and experience of pain leads easily to obscurity of thought upon this subject. But experience does not, if we may trust the general law of evolution from simple to complex, at the first contain consciousness of experience. This latter element is but gradually built up into experience, though in the end they are so permanently united in developed ego life that it is difficult to perceive their distinctness and independence. That pain and pleasure are cognized as facts of consciousness seems to us clear, but this does not deny that for us, at least, they may be cognizable only in fusion with other elements, as with sensation or volition. But whether known only with other elements or not, pleasure-pain is equally known only by direct introspection. I know directly and immediately pain and pleasure when I experience them, though they always occur bound up with some sensation. It may be that I never experience mere pain but some kind of pain, as a pricking pain, burning pain, etc., and that I always recall pain by its sensation tone, that I cannot isolate it by any act of attention. (E. B. Titchener, Philosophical Review, vol. iii., p. 431.) However I know that I have pain as well as I know that I have a pricking or burning sensation. “Did you feel the prick?” “Yes.” “Was it painful or pleasurable?” “Pleasurable”; such a common colloquy implies as direct consciousness of the pleasure-pain as of the sensation. That I can at once discriminate a sensation as either pleasurable or painful certainly shows a direct awareness of pleasure-pain. If pure pleasure-pain is primitive consciousness (see chap. ii.), it must be most rare phenomenon in such an advanced consciousness as that of the human adult: and it is not surprising that one should
- 55. search for itin vain. But in any case it could not yield to attention. Attention as cognition views its object in relation, in a milieu; it can reproduce only by fastening upon something to reproduce by, but pure pleasure-pain has nothing connected with it. Again, attention as volition cannot reproduce mere pleasure-pain which is not volitional in its origin and growth like sensing, perceiving, or ideating. We merely “suffer” pain. Both pleasure and pain in themselves are purely passive; willing cannot directly affect them, and they are not, like cognitions, modes of volition, or effortful activities. For man to have a primitive consciousness by exercise of will would be quite as difficult as to turn himself into a protozoön. Further, would not attention as introspective alertness to discover such a fact of consciousness as pure pleasure-pain denote that consciousness is thereby raised far above the level at which such a phenomenon can occur? In general also constant introspective attention tends to defeat itself. A continual intentness and watching for a given psychic phenomenon is a state which, the more intense and persistent it is, tends to bar out the particular state watched for, and, indeed, all other states than itself. If attention as act engrosses, it defeats itself. If, however, undifferentiated pleasure-pain should at any time occur in human consciousness, might we become immediately and spontaneously aware of it? By its very nature it may escape conscious attentive investigation, but may there not be a direct and simple awareness or apperception of it? We might suppose that one man tells another, “I was very sick, and in state of coma I had pain, merely pain, not any kind of pain or pain anywhere, but just pain, that was all the consciousness I had.” Such an expression is intelligible, and may be a fact. However, it is in the phenomena of lapse and rise of consciousness that we see evidences that undifferentiated feeling probably occurs, and that sometimes in high psychisms. In the following chapter we discuss then this point as a matter of judgment of tendencies, rather than on basis of direct evidence of introspection, though this is not barred out.
- 56. CHAPTER II ON PRIMITIVECONSCIOUSNESS Science views the world as an assemblage of objects having mutual relations. In this cosmos of interacting elements certain objects become endowed with mental powers by which they accomplish self- conservation. Just what these objects are and how they attain mental quality is beyond our direct investigation. However, assuming consciousness as a purely biological function, as a mode for securing favourable reactions, we can discuss the probable course of its evolution under the law of self-conservation. Mind, like all other vital function, must originate in some very simple and elementary form as demanded at some critical moment for the preservation of the organism. It is tolerably obvious that this could not be any objective consciousness, any cognitive act, like pure sensation, for this has no immediate value for life. It was not as awareness of object or in any discriminating activity that mind originated, for mere apprehension would not serve the being more than the property of reflection the mirror. The demand of the organism is for that which will accomplish immediate movement to the place of safety. The stone pressed upon by a heavy weight does not react at once to secure itself, but is crushed out of its identity; but the organism reacts at once through pain. It is certainly more consonant with the general law of evolution that mind start thus in pure subjective act rather than in mere objective acts, like bits of presentation or a manifold of sense. We shall now endeavour to elucidate this conception of pure pain as primitive mind, first from the general point of view of the law of self- conservation, and secondly from particular inductive considerations. It is very difficult to conceive what this bare undifferentiated pain as original conscious act was, it being so foreign to our own mental acts.
- 57. Our psychoses havea certain connection one with the other, and a connection which is cognized as such, so that the whole of mental life is pervaded by an ego-sense. But primitive consciousness must have been by intermittent and isolated flashes. The primitive pain, moreover, was not a pain in any particular kind, but wholly undifferentiated or bare pain. There was no sense of the painful, but only pure pain. Nor was there any consciousness of the pain, any knowledge or apperception of it. The pain stands alone and entirely by itself, and constituting by itself a genus. Now to assert that this general pain exists, is not, of course, realism. The pain is a particular act, though it is wholly without particular quality. It is not a pain as one of a kind distinct from other kinds, but it is comparable to a formless, unorganized mass of protoplasm which has in it potency of future development. Pain may exist as such, but not a consciousness or a feeling. It is meaningless to say that the first psychosis may have been a consciousness in general form which was neither a feeling, a will, or a cognition, but the undifferentiated basis of these, nor can a feeling per se exist. The expressions, painful consciousness, and painful feeling are deceptive; there is no consciousness which pains, but consciousness is the pain, and the feeling is not pleasurable or painful, but is the pleasure or pain. “Feeling,” as I have said (Mind, vol. xiii., p. 244), “has no independent being apart from the attributes which in common usage are attached to it, nor is there any general act of consciousness with which these properties are to be connected.” Further, the law of conservation requires us to associate with this primitive act of blind, formless pain the will act of struggle and effort which is as simple and undifferentiated as the feeling. And these two we must mark as the original elements of all mental life. Strenuousness through and by pain is primal and is simplest force which can conduce to self-preservation. It is thus that active beings with a value in and for themselves are constituted. The earliest conscious response to outward things is purely central and has no cognitive value. The first consciousness was a flash of pain, of small intensity, yet sufficient to awaken struggle and preserve life.
- 58. Pleasure, then, wehave excluded from playing any rôle in absolutely primitive consciousness. Pleasure and pain could not both be primitive functions, and of the two pain is fundamental in that the earliest function of consciousness must be purely monitory. Pain alone fulfils primitive demands, and secures struggle which ends in the abatement of pain through change of environment or otherwise. Pain lessens, but pleasure does not come, but unconsciousness instead, for no continuous organic psychic life is yet evolved. As long as pain continues there is effort and self-conserving action; when pain ceases, consciousness ceases, because the need for it is gone. Each fit of pain subsides into unconsciousness as struggle succeeds, and there is no room for even the pleasure of relief, which, indeed, must be accounted a tolerably late feeling. As far as the lowest organisms have a conscious life it is a pain life, but they have a Nirvana in a real unconsciousness. The evolution of pleasure must be accounted a distinct problem. The law of evolution is, that origin of function and all progressive modification arise at critical stages. Thus it is in painful circumstances that the origin of mind is to be traced, and the important steps in its development have been achieved in severest struggle and acutest pain at critical periods. Pleasure is not then the original stimulant of will, but is a secondary form. Pleasure has an obvious utility which is far from the absolutely primitive. The pleasure-mode early enters, however, to sharpen by contrast the pain-mode, and it is only by their interaction that any high grade of psychic life could be built up. The development of pleasure cannot be from pain, but as a polar opposite to it. We cannot bring the development of mind into a perfectly continuous evolution from a single germ, as is the case in biological evolution. In a sense we may say that pleasure and pain are complementary, like positive and negative electricity, but the comparison cannot be pressed. We cannot, indeed, carry it so far as to believe either absolutely essential to the other. We mention, then, the evolution of pleasure as a problem which is yet to be dealt with in full. However, that it is not original element in mind is easily seen from this. As we ascend the grades of psychic life the pleasure-pain gamut
- 59. lengthens, and aswe descend, it shortens, with pleasure always as the intermediate factor. Thus, if we can represent it by a line, Pain Pleasure Pain ───────────┼───────────┼─────────── any single element which can affect psychic life, as temperature, moves through a highest pain intensity, an intermediate region, then to pain again as effects in a range from a very high temperature to very low, or vice versâ. Now, this gamut in a human being, from the intensest agony from heat to the greatest suffering from cold, consists of very many notes, but the step to unconsciousness is always at one end of the scale. In lower psychic life it shortens, but always at the intermediate points where pain merges into pleasure and pleasure into pain, and thus in the lowest form the original element of consciousness as feeling is seen when only the two extremes remain, namely, primitive consciousness as pain reaction. As the step from feeling—consciousness to unconsciousness is through a pain, this certainly points to pain as the original feeling, and the first element of consciousness. We must suppose then that the first organism which attained consciousness felt pain, that if this came from temperature, for example, that intense heat and intense cold would both produce a pain one and the same in nature, bare pain, not sensation of heat or cold. And this pain-consciousness response came at first only at the application of these critical temperatures, all other degrees not bringing any response. If consciousness like other functions originated as an infinitesimal germ at some crisis in life, it must have been with pain. The pleasure function, unlike the pain, does not originate in life and death crises. That pleasure is secondary is also suggested by this, that pleasure is mainly connected with such late formations as the special senses, whereas pain is prominent with earlier functions. Thus we have pleasures of taste, but visceral pleasure is scarcely noticeable, though visceral pain, as colic, may be very acute. Wild animals, which feed often under fear of interruption or in extreme hunger, bolt their food without tasting, and so miss taste pleasure, and this seems to be the type of primitive feeding.
- 60. The origin ofpleasure is then, I think, to be traced as an intermediary feeling between pain as produced by excess, and pain from lack as differentiated form. Pain as original and undifferentiated is the same whether resulting from excess or lack, but it is only after it has differentiated so far as to be in two modes that pleasure can enter as a mediate form of feeling and become a directing force to advantageous action. The primitive pleasure-pain gamut was this: Lack Pain Pure Pleasure Excess Pain ──────────────┼──────────────┼────────────── A general survey from the point of view of self-conservation leads us then to regard the original psychic state as a pain-effort form. There is first a purely undifferentiated sense of pain and closely consequent a purely undifferentiated nisus. There is neither sense of objectivity in general, nor in any special mode, nor is there feeling of pleasure. And the study of what seem to be the earliest forms of mental life in the child and in the lower animals points toward this conclusion. Preyer, in his studies on the mind of the child, expresses his conviction that the feelings “are the first of all psychical events to appear with definiteness,” and that at first in no manifold forms. He adds, “The first period of human life belongs to the least agreeable, inasmuch as not only the number of enjoyments is small, but the capacity for enjoyment is small likewise, and the unpleasant feelings predominate until sleep interrupts them” (Mind of the Child, Part I., New York, 1888, p. 143, cf. p. 185). Since in the embryology of the mind as in that of the body the individual repeats in condensed manner the evolution of life, we judge that these observations point toward the genesis of consciousness in a single feeling state, pure undifferentiated pain. The earliest consciousness we can discover seems to approach this type. The close observer of very young infants must feel that the meagre psychic life they may have consists mainly of intermittent pains interrupted by comparatively long periods of unconsciousness in sleep. Of course, the earliest psychic life of the infant is not absolutely primitive both on account of heredity and on account of pre-natal experience; but in its general form it, no doubt, reverts toward the original status of mind. This original state, to which that of a very young infant is akin, was merely pain, which knew not
- 61. itself nor itsrelation to other states, nor its relation to the external world, but was a wholly central subjective fact, and so was expressed only in wild and blind general movements. The very lowest types of psychic life which we can interpret seems to feel and nothing more. They do not feel at anything, and do not feel because they know, nor do they have definite kinds of feeling. Pure feeling as bare pain and as undifferentiated pleasure is certainly far removed from our ordinary conscious experience, yet it may sometimes appear in a survival form, especially in sluggish states, in waking from sleep, and in recovering from anæsthetics. We are sometimes awakened by a dull pain which was evidently in its inception mere bare pain without differentiation. But in all such cases the pure pain or pure pleasure is but momentary, and is quickly swallowed up in a flood of manifold sensations. Many objects by many modes of sense at once invade and possess consciousness, and the early indefinite mode vanishes so quickly that we very rarely have time to note it by reflective consciousness. But it is not merely in exceptional states of developed consciousness that we may trace the elementary form of feeling, but we may believe it to be fundamental to consciousness in general. It is natural for us who are so pervaded and dominated by sense of objectivity to see in it the causal element in mentality; feeling and will seem consequent to it, and we apprehend and feel accordingly. But the order of evolution was not from knowledge in any form to feeling, but the reverse, and we may suspect that in the completest analysis consciousness will still be found to obey its original law. If the rise of knowledge was at the instance of feeling, it is certainly unlikely that a fundamental order should be more than apparently reversed. The order of consciousness is really the reverse of the order conceived by the objectifying consciousness, and this is a point where cognition by its very nature as objective may be said to obscure itself. To apprehend is to bring into relation, and the relation is very easily attributed to what is purely unrelated, to pure subjectivity. Thus here in the interpretation of merely subjective facts knowledge tends to stand in its own way. It is only objectively that the objectifying can appear causative of feeling; subjectively sense of object must always
- 62. be taken assubsequent to a pleasure-pain psychosis. The object communicates or causes the feeling, but the subjective order is as such of necessity the opposite; the object does not come in view; there is no relating, until feeling has incited to it, and gradually the mind reaches out to an objective order from the purely central fact. In every psychical reaction there must be the purely central disturbance before the rebound to the actuality occasioning the disturbance. I must feel before I can discriminate or have any sense of the communication of the feeling. This means that when external objects are brought into relation with a wholly unanticipating consciousness, the first element in psychosis is always pure pleasure or pure pain. Thus, on a cold, dark day a sudden rush of sunlight on a blindfold man causes pleasure, then feeling warm, and then sense of warming object. The glow of pleasure and the pang of pain merely as such is in all cases precedent to any objective reference. Pure centrality of response, I thus take to be the initial element of all psychosis, primitive or developed. The first tendency in every consciousness is pure pain-pleasure, complete subjectivity which, however, in higher consciousness is so quickly lost through practically consentaneous differentiation that all traces of it seem wholly extinguished. Pure subjectivity must be pronounced the most evanescent of all characters in developed minds and yet the most constant. It is the inevitable precedent in every sensation and in every perception. We always experience pleasure or pain before the pleasurable or painful. A bright colour gives pleasure before we see it, and this pleasure incites to the seeing it. But so fully has the objective order been wrought into consciousness as a mode of interpretation that the great majority on reading the preceding sentence will mentally at first attribute sense of objectivity from the expression “bright colour gives pleasure,” as if there were pleasure at colour, a colour-pleasure, whereas is meant pleasure and nothing more,—bare, undifferentiated pleasure. The objective statement, however true, is no measure of subjective fact, but this twisting of subjective fact to correspond with objective order is so embedded in language and common thought that it will perhaps always remain the form of ordinary thinking, like common- sense realism and geocentric appearance. The expressions, it pleased
- 63. me, it painedme, and the common modes of speech in general, are fundamentally misleading. Pleasure and pain bring their objects, not objects pleasures and pains. Pleasure per se does not come for and in consciousness from the object,—though this is objective order—but the object for and in consciousness comes from the pleasure. Pleasure and pain always precede any cognizance of the thing, and it is only the combination of the two elements that constitutes pleasure or pain of or at a thing. The primitive element, the original feeling movement, also excludes subject as real object; both the “it” and “me” are not yet apparent; there is not yet identification of experience with subject or object, and in fact no sense of experience at all. The psychologist must retain common expressions, however, but, like the astronomer who retains such phrases as the sun rises, the sun sets, he must reverse common interpretation and correct natural error. Guided by this principle we note an obvious error in the interpretation of child consciousness. If a bright-coloured object is passed before the eyes of a young infant we may conclude from its expression that a pleasure-consciousness is awakened, but we are probably quite at fault if we conceive it to have a consciousness of bright, and that this consciousness preceded and gave rise to pleasure and gave it a quale as pleasure-brightness. Sense of pleasure-object is manifested by appropriative activities, but in the very young, where these activities are lacking, the response to object is best regarded not as in any wise sense of object, nor even any kind of sensation, but as a pure subjectivity of pleasure. Of course the same remarks apply to the pain side of the child’s experience. The purely subjective experience, while it becomes more and more evanescent factor as mind develops, yet always maintains its place as the initial point and vanishing-point of every psychosis. Every psychosis beyond the most primitive must be accounted a feeling-will- knowing group. These psychic forces exist in a correlated union generally comparable with the correlated activity of physical forces like electricity and heat. Each psychosis repeats in itself, in tendency form at least, the essential stages in the evolution of consciousness. Every psychosis rises from the pure pleasure-pain as the lowest level of mentality like a wave, and like a wave falls back into it again. Every
- 64. wave of consciousness,whether it rises slowly or rapidly, whether it subsides gradually or violently, rises from pure subjectivity and comes back to it again. This absolutely simple feeling phase is accomplished so rapidly in ordinary human consciousness as to be rarely perceptible, but in lower consciousness it often exists as mood, as more or less permanent psychosis. The Brahmans attain artificially a subjectivity akin to this through their expertness in mental control and manipulation. They succeed in reducing and keeping consciousness in some very simple type, and their Nirvana may be considered as a state of pure subjectivity on the pleasure side. They, of course, cannot really attain this state or, at least, keep it, for pleasure is at bottom relative, yet they come to something approaching it. Pain at its height just before unconsciousness is reached, is always of the pure subjective type. In slow torture pain increases to a maximum intensity in pure pain, beyond which there is a gradual loss of intensity and consciousness in general, till ultimate failure of all consciousness. From the maximum intensity on to the end, consciousness is entirely subjective. Pleasure at its maximum attains only comparative subjectivity. Such facts tend toward a theory of mind which makes its original and fundamental act purely central; mind starts as in a germ which pushes outward till it penetrates space and time, but not in any reverse motion a pushing inward of a series of presentation forms. We shall now notice certain of Mr. James Ward’s statements on primordial mind—in the article Psychology, Encyclopædia Britannica— in which he controverts feeling as original and simplest unit in mentality. Mr. Ward regards “the simplest form of psychical life” as involving “qualitatively distinguishable presentations which are the occasions of the feeling.” Presentation is primitive and initial in all consciousness, and cognition—feeling—will is the order for all mind. We always act as we are pleased or pained with the “changes in our sensations, thoughts, or circumstances” of which we are aware. Some presentation form is, throughout all our experience, the precursor and cause of feeling, and feeling can never be said to exist in a pure state as bare pleasure and pain totally without cognitive value. On the contrary, I conclude from general considerations and from special indications in our own minds that pure pain is the original
- 65. element, and thatpure pleasure and pain are fundamental in all mind. Pure feeling arises from objects, indeed, but is still wholly unknowing of object and without qualitative aspect. Pure feeling is the constant incentive to all knowing and will activity. To say that I am pleased with a thing is to transform objective order into subjective fact. Pleasures and pains certainly come from things but this does not invariably rouse cognition of them as so coming, or of object as causative agent. The governing and essential fact of mind is always pure feeling, which, by reason of its perfect centrality, necessarily and naturally tends to elude observation. Every act of consciousness begins and ends with pure feeling, but mind, as far as it minds itself, is most apt to see only culminating phases rather than the obscure and inner forces which constituted long outgrown stages. The prominent facts of late consciousness are always very complex. Cognition as revealer unites with the known and inevitably, but strongly tends to regard itself as the determining and causative agent, whereas by its essence and function it is secondary. Cognition does not create its object, except in the view of a transcendental philosophy. Mr. Ward asserts that phenomena of pleasure and pain involve change in consciousness with consciousness of change whereby we are pleased or pained. A changing presentation continuum is impressed upon mind, and it is by awareness of these changes that feelings are caused. This is certainly a complex mode to be assigned to all consciousness. This asserts that primarily consciousness merely happens in presentation form as determined from without, but I take it that the evolution of faculty is always acquirement, not mind determined, but mind determining, achieving its own growth in blind struggle. Mind is wholly an inward growth, not a series of givens; and presentations are accomplished not merely in it but by it. The fundamental principle is that while objects do determine conscious functions, it is only through self-conservative interest, through pleasure and pain reacting to them. All sensations, intuitions, presentations, are at bottom achievements as forced by law of struggle for existence. They do, indeed, seem to come of necessity and spontaneously to adult human consciousness, but developed faculty by virtue of being such does not have to attain beginnings.
- 66. But we notealso this, that while all consciousness is change in the sense of being dynamic, of being an activity, this does not include consciousness of change. Consciousness as a changing factor is very distinct from consciousness of that change, and does not necessarily include or imply it. That the forms of activity which we group under the general term consciousness have their existence wholly in movement and change is true, but this does not necessitate that the changing elements should be aware of the change as such. Different things may be felt and known, but this does not always result in being known as different. This brings in comparison, consciousness of relation, which is certainly beyond primitive consciousness. In early mind we conceive that new elements are continually taking the place of the old, that change is incessant, yet without sense of the change. So far as the earliest consciousness is spasmodic and intermittent, appearing in isolated flashes, we cannot speak even of change in consciousness, much less of consciousness of change, for there is no continuous thread, no integration, consequently change is not in consciousness from a consciousness to a consciousness, but the only change is from a consciousness to unconsciousness. In the whole life of some organisms we may believe that only three or four pains or pleasures occur, entirely subjective and undifferentiated, and this collection of consciousnesses where state does not follow and influence state, where there is no complexity, is scarcely to be termed a consciousness which changes, much less that is aware of change. It is not improbable that even with civilized and educated men mind may sometimes lapse so far that changes occur with no awareness of change. In such sluggish conditions as when half asleep we may experience succession of consciousnesses without noting succession, each phase standing alone in itself and by itself. While consciousness is maintained as consciousness—that is, a continuance of conscious states—by the change, it is obviously not necessary to this that there should be awareness of change. Here as elsewhere we must keep clear of the mistake of making consciousness more than a general term for a group of phenomena. Consciousness as such has no reality or existence, but merely denominates a sum of consciousnesses. The phrase, change of consciousness, and similar expressions easily
- 67. convey the impressionthat consciousness is a changing something. But we know that consciousness does not exist as a general indefinite something which changes or has other properties, but is merely a name for certain activities and functions. The formula of Mr. Ward’s hardly applies to developed consciousness, much less to undeveloped. Consciousness even in man cannot be regarded as a something which changes in sensation and presentation forms as pure givens, determined with immediate completeness from without, and these changes perceived, and pleasure and pain result. On the contrary the immediateness and spontaneity of presentation forms in our ordinary adult human consciousness are in appearance only; they stand first before us because they have reached a dominance through heredity and education, but still the latent and inward order is always from feeling to knowledge and not vice versâ. The accomplishment of presentation is usually so marvellously rapid in perceptive beings, and acts upon such slight incentive that it is only under very rare conditions of regression, or when developing a new sense or new form of sense that we see that the moving element in mentality is pure feeling. Thus, for example, in being awakened from sound sleep by a bright light suddenly brought into the room, the order of consciousness is, pure feeling of pain, sensation of light, perception of lighted object, and not the reverse; whenever we can catch consciousness gradually awakening we can always identify this order. The lighted lamp, objectively speaking, certainly caused the feeling of discomfort with which consciousness began, and this feeling roused the mind to both sensation of light and perception of lamp. I, of course, have a feeling as to the visible object only after seeing it, but this is altogether distinct from the feeling which incites to the seeing. A vague, undifferentiated pain or pleasure is always initiative, but pure pleasure-pain is often so low in intensity that it does not start any cognitive act. In a general way the influence of feeling and emotion upon cognitive act in higher psychical life is acknowledged by common observation. The wish is father to the thought—we see what we want to see. What we observe depends upon prepossession, interest, and
- 68. the whole pleasure-paintone. The mind must be determined to cognitive act by interest of some kind, and even for advanced consciousness with all its strength of inherited aptitude total loss of interest ultimately leads to loss of perceptive power. The impetus of all previous cognitive effort will carry on cognition, of any high order, at least, but a comparatively short time. Blot feeling out of life and all nature would soon become a dumb show and quickly fade into nothingness. Absolute passionless receptivity is impossible under the conditions of reality, and pure presentation forms never come as antecedent and causative to feeling. We have constantly to bear in mind that in the nature of the case the simplest elements and fundamental laws are hidden and certainly far from conspicuous in highly developed mind, which is an intricate nexus of feeling, will, and cognition constantly acting and reacting on each other. As a general statement, then, impliedly as to mind in general, and implicitly as to the developed human mind, the proposition that consciousness is fundamentally aware of changes in itself as the basis and cause of all feeling is an assertion which may well be questioned. Certain it is that being “pleased or pained with the change” is not feeling in general, but a particular kind of feeling, namely, feeling of variety and novelty. Further, to be pleased with a thing for itself alone is not to be referred to pleasure or pain “with the change.” There is intrinsic pleasurableness and painfulness which does not come under the head of pleasure or pain of change. From both an a priori point of view of the law of self-conservation, and also from a brief survey of certain forms in comparative and human psychology, we incline towards accepting pure pain as the original consciousness which is very soon differentiated into excess and lack pain with evolution of pure pleasure. Will exists throughout as incited by feeling. Much, indeed, is to be done before this theory of the nature of mind is either fully elucidated or proved; but I believe that the assumption of mind as life function leads toward such a theory. Sensationalism and intuitionalism are both mistaken as to the origin and essence of mentality. Consciousness is not at bottom any mode of cognition, either as more or less freely accomplished by a “mind,” or as more or less mechanical impression from “things,” but it is primitively and
- 69. fundamentally pain andpleasure as serving the organism in the struggle for existence. It is strange that evolutionary psychologists have so generally missed this point of view, and maintain sensationalism. Comte, indeed, acutely remarks (Positive Philosophy, vol. 1, p. 463) that “daily experience shows that the affections, the propensities, the passions, are the great springs of human life; and that, so far from resulting from intelligence, their spontaneous and independent impulse is indispensable to the first awakening and continuous development of the various intellectual faculties.” He here assumes the introspection which he elsewhere denies as psychological method, and enunciates an important principle which he never carried out. Horwicz has made a survey of feeling as fundamental aspect of mind, but his discussion is physiological. Our conclusions have been founded on general considerations and on the phenomena of growth of mind in general and particular. Another line of evidence would be decadent mind. Mental powers should decline and vanish in the reverse of the general order in which they arose; the order of disappearance should be the reverse of appearance, and if pain-pleasure be primitive, we should expect to find it both the first conscious element in infancy and the last in old age. The last stage of senility seems sensitive only to organic pleasures and pains. Further, old age does not so much seek pleasure as guard against pains, and this fact is in line with our treatment of pain as prior to pleasure and more fundamental than it. We may consider it likely that conscious life in the individual begins with a pain and ends with a pain. Senile psychology on this and other points is worthy of far more attention than it has received, for it is on the whole more accessible and trustworthy than infant psychology. With regard to Mr. H. R. Marshall’s remarks (Philosophical Review, vol. 1, p. 632), it is sufficient to say that I lay no great emphasis on either pain or pleasure being the first fact of consciousness; but my main contention is that the primitive facts of consciousness are of the pain-pleasure type. While I have noticed some considerations as implying pain to be the first consciousness phenomenon, yet I am satisfied that pain and pleasure are correlative and complementary,
- 70. each implying theother. Further, I do not regard pain as “primal sense,” but as primal fact. Pain is not in any wise a sense, and sense of pain can only mean capacity for pain, or actual pain experience. Again, I do not, as Mr. Marshall implies, regard pain as the differentiating basis of subsequent evolution, but rather as mere prius and impetus, and hence I do not look for pain-pleasure to disappear with mental evolution, nor yet to mark divisions in “sensational phenomena”; but it will ever remain in representative forms, at least, as increasingly complex stimulant of all mental life. The objection urged by Höffding and others to the primitive nature of pure feeling is that we sense before we feel pain or pleasure; thus we have the sensation of touch before we feel the pain from contact with a hot stove; we feel the pin, then the pricking sensation, then the pain. This precedence has been measured by Beau and others. But what is the significance of these well-recognised facts? Do they show that pain-pleasure originates always in sensation? What is the origin of tactile power? How and why was the first tactile effort made, if not at impulse of some pain-pleasure? When conscious life was at pre-tactile stage—before it had learned to touch—it had no pain from touch, but it had pain. We can scarcely deny that a pre-tactile stage exists, that all sensation was originally a sensing—an exertive act, that it did not come, but was attained; for all the growth of sensitive power in the race proceeds thus at present, and the law of present psychic development in this regard seems general. But it is pain- pleasure which forces all action; here is the impulse which brings exertion whether as sensing or otherwise. A doctrine of spontaneity is against the general law of development by struggle. It is certainly true that, standing with my back to the stove and inadvertently coming in contact, I, without any previous pain-pleasure impulse and without exertion, have sense of touch, then pain. But this spontaneity is not original factor; it is the result of inherited powers. When tactility has become a well-developed power and is handed down to descendants, then contact with things is immediately and spontaneously realized in the form of touch, which contact would originally have been unnoticed. That is, the severest condition—a red hot stove—would impress the lowest psychism only in terms of mere pain, and so result
- 71. in general reactionsof minimum service. The early psychism which is just in process of achieving sense of touch would have pain, and then with effort touch the object and thus attain some more special reaction of more particular service. But the tactile, like all sensing activity is anticipatory, it is a finder, an interpreter. Suppose I bring a very fine needle toward your eye, you may see it and avoid it; but suppose your eyes are shut the eye comes in contact with the needle, and you have sensation of touch; but you are sound asleep, then pricking sensation may wake you as needle proceeds deeper, but in profoundest sleep undefined pain may be the first consciousness to result. Now the needle might be so small as to be seen with great difficulty by the waking man, or invisible, or to be touched with great difficulty; but this stage of exertive action for the sense is only relative, and in the history of mind the very grossest forms were at one time only dimly seen by intensest effort, and lower still, touched only by intensest effort. Seeing originated in looking, and passive touch in active touch, as moved by interest or direct pleasure-pain. Now pain is not in the mere sight or touch, but is suggested by them. The whole order—seeing, touching, feeling prick, feeling pain—is the reverse of evolution order. The rational mode, then, of interpreting the origin of any sense, whether tactile, visual or other, is not by receptivity, but through struggle at critical stage when great pain is actual or imminent. Thus, if the conditions of life required the development of a special sense of magnetism, it would surely arise by strongest effort, as, indeed, all progress in special sensitiveness is now being accomplished. Thus, the anticipatory and premonitory function of sense does not make it original, rather the contrary; it is guide and significant of pain-pleasure. It is obvious that the cognitive tendency once established becomes an instinct of objectivity and governs the whole mentality. This is obviously the case with man. He does not exist in that sluggishness and semi-consciousness where pain-pleasure must arise as primitive impulse, but by habit and instinct he is passively and actively cognitive. The eye is continually seeing things spontaneously, the hand touching, but as to some very small object we have to exert effort to see or touch, and this was undoubtedly the mode by which
- 72. all seeing andtouching arose. It is because generations of ancestors actively sensed, that we automatically sense; the tendency has become ingrained in mind. So it is that man is predominantly sensing, is continually and naturally awake to objective conditions, is constantly anticipatory, and so normally senses before he feels pain-pleasure. However, a man in a “brown study,” inadvertently touching a hot stove, has pain, then warmth, then touch sensation, and actively realizes these. So in deep slumber mentality often begins with pain- pleasure. At bottom the reason we have pain from a sensing is because we had originally pain-impulse to that sensing, and the pain therewith. Thus tactility, arising as effortful sensing, was produced by pain from thing to be touched, to be sensed in its experimental value. By innumerable painful experiences with hot things, the hot thing is tactilily appreciated; and as touching is actively pursued by organism on the alert, the associated pain is more and more quickly realized from given object. In origin pain was felt from the hot thing in contact, before either sense of warmth or contact was sensed; it was this pain that forced to sensing and development of cognition, which, however, ultimately became habit, and things were constantly appreciated and anticipated. Thus the touch-warmth-pain order is established. Sense is significant of pain-pleasure, but the pain- pleasure came not at first from the sensing, but the contrary; sensing was determined by it, and became correlated with it, and became sign of it. The progress is from initial subjectivity to an instinctive constant objectivity. This objectivity is reflected in all objective expression as language; “the heat was painful,” “it hurt”; the “it” being tactual thing, etc., etc. However, if we look for primitive consciousness, we must find it only in primitive organisms in their primitive stage, and in man most rarely only as tendency in profound relapse. We must mark this, that cognition is not to be evolved out of feeling, but at instance of feeling as impelling the knowing effort or volition. We may suppose that primitive consciousness still exists in the lowest types of life, but it may also be the sub-consciousness in the higher types. Viewed biologically, what is sub-consciousness? The earliest living aggregations attain but a very slight degree of common life, and very slowly do the cells, under the pressure of
- 73. serviceability in thestruggle for existence, give up their independency and become interdependent, each thereby giving up some functioning to be done for it by others, and in turn functioning for others. Thus it is but slowly that a stomach is specialised, the cells in general in the organism long retaining and exercising some digestive function, which is properly termed sub-digestion. In this way a soup bath gives nourishment. If psychic function specializes gradually like other functions, we shall have in the same way a sub-form here, a sub- consciousness which stands for lower centres, and not for the whole organism as such. The wider, higher, and more specialized psychic centre does not at once extinguish the lower. Now what is a high organism but an involved series of combinations of combinations? With every new integration a higher plane is achieved, and the vital process has a wider functioning: but the physical or psychical activity so far as it does not pass over into the service of the new and higher whole remains as sub-function. With every new stage in evolution the integrating psychic factors only partially lose themselves in effecting a common psychism for the new whole, a sub-consciousness and a sub-sub-consciousness, etc., are still carried on in survival. In man, physiologically speaking, it is the brain consciousness which is general. But we need not suppose this to extinguish all the lower ganglionic consciousness from which and by which it arose. If psychic function be correlative with other function, we must expect in man a vast amount of survival sub-mentality which, while not the mind of the man, is yet mind in the man. The individual knows necessarily only the general consciousness, for this only is his consciousness and constitutes his individuality, yet the doctrine of evolution would call for a vast deal of undiscoverable simple consciousness which never rises to the level of the whole organism’s consciousness. A cell or a group of cells may be in pain and yet there be no pain in the individual’s consciousness, and so unknown to this general consciousness. We have intimated that primitive consciousness may occur in a sub- conscious way in the highest organisms. But can this sub- consciousness ever be more than mere survival in its nature? or may it play essential part as basis of higher manifestations? If the
- 74. integration of mentalityis like other integration,—e.g. material which is based on molecular and atomic activity—it will be bound up in the activity of psychic units, which can be none other than sub- consciousness. That is, any common or general consciousness when looked at from below, and analytically is the dynamic organic whole of elements; it is a product of activities which are on another plane from itself. Roughly illustrated, I may say that my finger feels pain before I do. We conceive that at a certain intensity a sub-consciousness tends to rouse a general consciousness, and for a time maintain it; and losing intensity, the general consciousness disappears leaving only the sub-consciousness, which may long outlast the general form. Sub-consciousness, whether as survival or basal, is put beyond our direct observation, but it remains a necessary biological and psychological hypothesis. Here is exemplified for psychosis that law of the aggregation of units in hierarchical order, that wheel within wheel structure of the universe, upon which I have touched in Mind, ix. pp. 272-3.
- 75. CHAPTER III THEORIES OFPLEASURE-PAIN The bearing of our studies on a theory of the conditions of pleasure-pain is obvious. If we consider pure feeling as the primary, fundamental, and conditioning mentality, it stands before all other mentality, and cannot be interpreted as conditioned. Pain as primum mobile is not intrinsically dependent on any other psychosis. Hence we run counter to the Herbartian School, which maintains that psychism exists from the first for itself as intellectual ideational activity, and that pleasure-pain is but reflex of the efficiency and ease, or the inefficiency and difficulty of this activity. The checking of the current of ideas may give a pain, but our exposition has been that pain arose before ideas or presentations of any kind, and long before any interference could be felt as pain. Again, if we say “all pain comes from tension” (Mind, xii. p. 6), we have to ask, Tension of what? If we say tension of sensation or ideation, this is Herbartianism merely. How also can tension be felt as painful, except through sensation of tension, which is a feeling of intense sensation—obviously a late psychosis? And certainly pain is more than a general consciousness fatigue. And further stress and strain result in pain, because we imply these as painful activities by the very notion of the words. A stress or strain is assumedly painful activity, but this is not explanation. But apart from this, if the organism felt pain merely as direct result of struggling and straining, it would cease activity; activity and evolution would stop. It may be that by tension is not meant a mode of consciousness, but of nervous or muscular activity; but as we are now considering psychosis only as conditioning pure feeling, we leave this aspect for
- 76. discussion till alittle later. But on the psychical side, that all pain is a by-product of over-intense consciousness, intellectual or volitional, that the origin and development of pain is in a mental intensity which has gone beyond a certain point, this seems, on general evolutionary grounds, unlikely. Here, indeed, is merely a very particular and rather late mode of pain. And may not pains themselves attain an intensity which is itself painful? It must be acknowledged, however, that the whole doctrine as to consciousness intensity, its nature, reactions, laws, and measurements is very obscure. Again, as to the theory that pleasure-pain is reflex of quantity of consciousness, that pleasure results from mental expansion, pain from mental contraction, this must, like the intensity theory, be considered as putting a late and special form as covering all forms. Mentality here exists for itself, and conscious self-development—a very late mode—is presupposed. The promotion of large complete free consciousness, the sense of progress and of unimpeded mental activity, certainly conveys high joys to certain choice natures, but they do not touch the vast majority of even human minds, much less animal. With the stolid an expanding consciousness is painful. Consciousness only as conscious of itself, and as self-developing, reaches a pleasure or pain as a felt furtherance or hindrance of its own expansion. All reflex theories take us above the realm of simple consciousness acting directly for life, and this is the very form which seems commonest, and which appears to be full of passing pleasures and pains. That consciousness does react on itself in late phases is plain, but if consciousness, like other functions, has developed from the extremely simple to the extremely complex, this self-reaction cannot be regarded as primitive. Not till consciousness becomes integrated as a manifold organism do pleasure and pain become prominent as reflexes. We are not now looking for the functional value of pleasure and pain in mind itself as an independent whole; but regarding its functional quality and that of all mentality in life values, and here the functional meaning of such reflexes is secondary. In mind, as organic continuous whole, pleasure-pain is both resultant and excitant; it
- 77. stands related toan antecedent state and it is stimulant to following states. Its function is excitant and it is the starting point of all other mentality, both originally and in the later manifestation. The having pleasure-pain is what starts both motor and cognitive volition. It has, indeed, been maintained that while pleasure-pain is not a product or concomitant of some psychosis, as sensation, it is itself a sensation, a definite mode of sensibility. I have a pain sense just as I have a temperature sense, I feel pain in the same way as I feel warm, and by the analogous sensory nerves. With reference to this theory we must ask, since sensation is correspondent to modes of objects, to what mode is pain correspondent? Sense responds to modes of object, as light, and sonorous vibrations; but pain is not based on any such mode of objects. If pain were, there would have been long since a department of physics, which would have treated that basis just as it treats light, heat, sound, etc. But we all know that an object is not painful or pleasing in the same way that it is warm or cold, heavy or light. I do not say the stone feels heavy and painful, but I do say the stone feels painfully heavy, that is feeling pain is not a state of awareness. Further, having pain or pleasure is not by any sensing effort. I do not try to feel pain as I try to see the light of a star or feel the warm spot in a bar of iron. To be sure, the doctor asks his patient, “do you feel any pain?” and after a moment’s delay the answer may be, “yes,” but this is not in the nature of a sensing effort, but merely an attentiveness to bodily conditions as affecting mental state, not an objective attention but an analytical self-attention. Still further, a neural basis for pleasure- pain is altogether likely, but even if these nerves were found to be generally distributed over the body, this would not prove sensation, but merely that pleasure-pain is functional throughout the organism, diffusive organic consciousness. If pleasure-pain is primitive, and neurality and mentality correlate, the earliest nerve structure— ganglion—was a pleasure-pain organ. However, the sensory motor predominance is so early and complete that the current theory, as the more objective, is the natural physiologic interpretation. Again, it has been maintained that pleasure-pain is not a definite state of consciousness, but a quality like intensity, a modus which
- 78. must belong toall states. But if we assign pleasure-pain to such a category as intensity we must define just what we mean by this category. Is intensity a mere objective quality which we as observers assign to all psychosis, just as we do to electrical or luminous phenomena? or is it inherent element, an actual constituent, of every psychosis? If a man is angry and becomes more angry, intensity is increased; but we may conceive that he simply is more angry without being aware of this change of intensity, that is without every change of intensity being noted by consciousness. As introspection avers, it often happens that a man is both unconscious of his anger and unconscious of its increase. As I have frequently had occasion to note, simple natures are wholly unconscious of their emotions and of their intensity variations. That is, as matter of fact, intensity of feeling is not feeling of intensity. If you feel warm you feel differently than when you feel warmer, but this is no more than saying that when the iron is hot it is in a different state than when it is hotter. Intensity means the same in both cases. Consciousness, primitively, at least, is not self-awareness of its own changes in intensity. The feeling warm and the feeling warmer occur simply as facts which are subjectively unrelated and unmeasured by the consciousness which has the varying intensities. I strike a cow hard —result, intense pain; harder, more intense pain; this is correlative with, I strike iron, intense tremor; harder, more intense tremor. The cow experiences more intense pain, but does not consciously measure it off as such. I can say, “I feel hotter than I did,” but the cow does not appreciate and express its own sense of its experience. The language fallacy leads us astray. By our very use of terms, warm and warmer, and by our discussion of the matter, we imply a consciousness of intensity which is far from being primitive or general. It would probably be an overestimate to say that the intensity of one in a thousand psychoses makes itself felt as such in consciousness. That consciousness is not always conscious of its own intensity is then shown by direct introspection. And in general we must observe that every psychosis has its own intensity, which intensity may or may not be noted by a consciousness of intensity. If there come a
- 79. consciousness of intensity,this consciousness has its own intensity, which may be noted by a new consciousness, whose intensity may in like manner be noted by a new consciousness, etc., ad infinitum. That is, a consciousness is never its own intensity, and intensity is never a consciousness, such as pain or pleasure, but is mere comparative objective quality. Again, consciousness has almost from the first different degrees of activity, but it would be most unlikely that so complex an act as consciousness conscious of its own intensity should be primitive and early. Also, if consciousness develops as life factor it must be immediate utility which determines its early forms. Hence on this general principle of biologic evolution it is most unlikely that primitive organisms will both have consciousnesses and consciousness of their intensity, for of what direct and vital value is this intensity-consciousness as psychic mode? On the other hand it is obviously desirable that psychoses should early differentiate intensity as objective quality, i.e., without self-awareness of it, should have different degrees of a psychosis to meet different degrees of requirement; thus to fear strongly or weakly according to necessity of the case. To have fear set at one pitch for all cases is perhaps absolutely primitive, but differentiation is early. But to fear more or less, i.e., at different intensities, is not to have intensity as subjective element, an actual psychosis constituent appreciated as such, which is very late evolution since the demand for it is late. In thus defining the category of intensity we have plainly isolated it from the pleasure-pain category. We know pleasure or pain as act of consciousness just as we know volition or sensation. Pain and pleasure are definite facts like seeing or touching or willing, and are so recognised by common consciousness. One or the other may be involved in all experience, but this does not make them general qualities like intensity. Pain is a consciousness, intensity is not a consciousness. This is the immediate value of the terms, the very names convey distinctness of category. I have a pain, I do not have an intensity; I am in pain, I am not in intensity. My pain is intense, but I cannot say my intensity is painful. We experience pain and pleasure, but we never experience intensity.
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