Basic graph theory

BASIC GRAPH THEORY
P.JAYALAKSHMI
ASSISTANT PROFESSOR IN MATHEMATICS
SRI GVG VISALAKSHI COLLEGE FOR WOMEN(Autonomous)
Affiliated to Bharathiar University
An ISO 9001 - 2015 Certified Institution
Re-Accredited at 'A+' Grade by NAAC (Fourth Cycle)
UDUMALPET

GRAPHS, SUBGRAPHS AND
COMPONENTS
• Graphs
• Subgraphs and Some Special Graphs
• Graph Properties
• Paths ,Cycles and Components

Graphs
• The algebraic definition of a graph is given
as follows:
A graph G is a finite non-empty set V together
with a symmetric irreflexive binary relation A
on V. The elements of the set V are called the
vertices of the graph .The relation A is called
the adjacency relation.

Graphs
• The set-theoretic definition of a graph is
defined as follows:
A graph G is a pair (V,E) where V is a non-
empty set whose elements are called the
vertices of G and E is a subset of V whose
elements are called the edges of G.

Graphs
• The geometric flavour of a graph is given as
follows:
A graph G is a pair of disjoint sets V (where V
is non-empty) and E & a 1 -1 incidence
function f : E→ V.Elements of V are called
vertices of G and elements of E are called
edges of G.

Graphs
• The order of graph G is n = │V│ and size of
graph G is m = │E│.
• A graph of order n and size m is referred as
(n,m) graph.
• If an edge e corresponds to the vertex pair
(u,v) then e = uv ,that is the edge e joins the
vertices u and v.

Graphs
• Two graphs G = (V,E) and H =(U,F) are identical
or same or label isomorphic iff
V = U and for any pair u,v inV,uvєE iff uvєF.
• Two graphs G = (V,E) and H =(U,F) are
isomorphic iff there is a bijection ϕ : V →U such
that for any pair u,v in V , uvєE iff ϕ(u) ϕ(v) є
F.Then ϕ is called an isomorphism of G onto H.

Isomorphic Graphs
• An isomorphism of G onto itself is called
automorphism.
(i.e)It is a mapping from the vertices of the
given graph G back to vertices of such that
the resulting graph is isomorphic with G.

Graph Invariant
• A graph invariant is a function f from the set
of all graphs to any range of
values(numerical, vectorial or any other)
such that f takes the same value on
isomorphic graphs.When the range of
values is numerical(real,rational or integral)
the invariant is called a parameter.

Graph Invariant
• It is well known that we associate numbers to
mathematical objects in many ways.
For instance Determinant is associated to a
matrix,degree is associated to a
polynomial,dimension is associated to a
space,length is associated to a vector etc.,
There are several numbers associated with
graph.Such a number is called graph invariant.

Embedding of a Graph
• An embedding of a graph G on a surface S
is a diagram of G drawn on the surface
such that the Jordan arcs representing any
two edges of G do not intersect except at a
point representing a vertex of G.

Planar and non-planar Graph
• A graph is planar if it has an embedding on
the plane.
• A graph which has no embedding on the
plane is non-planar.
• A graph that can be embedded on a torus is
called a toroidal graph.

Multigraph
• A multigraph is a pair (V,E) where V is a
non-empty set of vertices and E is a
multiset of edges ,being a multisubset of V.
The number of times an edge e = uv occurs
in E is called the multiplicity of e and edges
with multiplicity greater than one are called
multiple edges.

Graph
• A general graph is a pair (V,E) where V is a
non-empty set of vertices and E is a multiset of
edges, being a multisubset of V ,the set of
unordered pairs of elements of V,not
necessarily distinct.
• An edge of the form e = uu(uєV) is called a
loop.
• An edge which is not a loop is called a proper
edge or link.

Graph
• The number of times edge e occurs is called
its multiplicity and proper edges with
multiplicity greater than one are called
multiple edges.
• Loops with multiplicity greater than one are
called multiple loops.

Underlying Graph of G
• The graph obtained by replacing all multiple
edges by single edge in a multigraph G, is
called underlying graph of G.
• Similarly if G is a general graph , the graph
H obtained by removing all its loops and by
replacing all multiple edges by single edges
is called the underlying graph of G.

Digraph
• A digraph D is a pair (V,A) where V is a non-
empty set whose elements are called
vertices and A is a subset of the set of
ordered pairs of the distinct elements of V
whose elements are called the arcs of D.

Multidigraph
• A multidigraph D is a pair (V,A) where V is a
non-empty set of vertices and A is a
multiset of arcs,being a multisubset of the
set of ordered pairs of distinct elements of
V. The number of times an arc occurs in D
is called its multiplicity and arcs with
multiplicity greater than one are called
multiple arcs of D.

General digraph
• A generaldigraph D is a pair (V,A) where V
is a non-empty set of vertices and A is a
multiset of arcs,being a multisubset of the
cartesian product of V with itself.An arc of
the form uu is called a loop of D and arcs
which are not loops are called proper arcs of
D

General digraph
• The number of times an arcs occurs is
called its multiplicity.
• A loop with multiplicity greater than one is
called a multiple loop.
• An arc (u,v)є A of a digraph will also be
denoted by uv,implying that it is directed
from u to v,u being the initial vertex and v
the terminal vertex.

General digraph
• If D = (V,A) is a digraph, then the graph
G = (V,E) where uvє E iff uv or vu or both
are in A, is called the underlying graph of D.
• If D = (V,A) is a general digraph, the digraph
obtained from D by removing all loops and
by replacing all multiple arcs by single arcs
is called the digraph underlying D.

Mixed graph
• A mixed graph G = (V, A ᴜ E) consists of a
non-empty set V of vertices, a set A of arcs
and a set E of edges such that if uv є E then
neither uv nor vu is in A.

SUBGRAPHS AND SOME SPECIAL
GRAPHS
• A subgraph of a graph G=(V,E) is a graph
H = (U,F) withU V and F E.
• A graph of order n with all possible edges i.e
when m = n(n-1)/2 is called complete graph
of order n and is denoted by .
 
nK

GRAPHS
• consists of a single vertex and is called
the trivial graph.
• has two vertices and a unique edge
joining them.
• is called as triangle.
1K
2K
3K

GRAPHS
• The graph has no vertex and no edge and
will not be called a graph.If the set V is empty
then is called as the null graph.
• A graph of order n with no edges is isomorphic
to any other graph of order n with no edges is
called empty graph of order n and is denoted
by
• Every graph of order n is a spanning subgraph
of
0K
0K
nK

nK

GRAPHS
• A graph G = (V,E) is said to be r-partite
(where r is a positive integer) if its vertex
can be partitioned as V = such
that uv is an edge of G then u is in some
and v is in some other that is,
everyone of the induced subgraphs is an
empty graph.
rVVV  ...21
iV jV
iV

GRAPHS
• If an r-partite graph has all possible edges,
that is uv є E for every u є and v є for all
pairs then it is called a complete
r-partite graph.
• A 2-partite graph is referred to as a
bipartite graph.
• The complete bipartite graph is called an
n-star or an n-claw.
jViV
ji VV ,
nK ,1

GRAPHS
• The complement of a graph
G = (V,E) has the same vertex set as G and
its edge set is the complement of E in V.
• A graph G is said to be self complementary
if G =
),( EVG 
E
G

GRAPHS
• Let G = (V,E) be a graph and F a subset of
the edge set E. Then the graph H = (V,E-F)
with the same vertex set as G and the edge
set E-F is said to be obtained from G by
removing the edges in the set F .It is
denoted by G – F. If F consists of a single
edge e of G,the graph obtained by removing
e is denoted by G – e.

GRAPHS
• A vertex v of a graph G which is not adjacent
with any other vertex of G is called an isolated
vertex of G.
• Let G = (V,E) be a graph and v be a vertex of G.
Let be the set of all edges of G incident with
v. Then the graph H = (V-{v},E- ) is said to be
obtained from G by the removal of the vertex v
and is denoted by G – v.
vE
vE

Basic graph theory

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