This content includes the basic terminology for graphs .Various kinds of subgraphs are dealt.

1. 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

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

3. 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.

4. 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.

5. 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.

6. Graphs

7. 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.

8. 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.

9. Isomorphic Graphs

10. 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.

11. Automorphism in Graphs

12. 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.

13. 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.

14. Graph Invariant

15. 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.

16. Embedding of a Graph

17. Embedding of a Graph

18. 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.

19. Planar and non-planar Graph

20. Toroidal Graph

21. 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.

22. Multigraph

23. 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.

24. 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.

25. 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.

26. Underlying Graph of G

27. 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.

28. 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.

29. 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

30. 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.

31. 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.

32. 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.

33. 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

34. SUBGRAPHS AND SOME SPECIAL
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

35. SUBGRAPHS AND SOME SPECIAL
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

36. SUBGRAPHS AND SOME SPECIAL
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

37. SUBGRAPHS AND SOME SPECIAL
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

38. SUBGRAPHS AND SOME SPECIAL
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

39. SUBGRAPHS AND SOME SPECIAL
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.

40. SUBGRAPHS AND SOME SPECIAL
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

41. THANK YOU