This document discusses isomorphism and connectivity in graphs. It defines isomorphism and lists important results about properties that must be the same for two graphs to be isomorphic, such as the number of edges and vertices. It also discusses the contrapositive of these results. The document then defines connectivity in graphs, including connected graphs, connected components, cut vertices, and vertex connectivity.

1. ISOMORPHISM
Mr. S.L. Khairnar

2. Isomorphism

3. Note. Two simple graphs that are not isomorphic are
called non isomorphic.

4. Important Results

5.  If G1 and G2 are isomorphic then number of
edges of G1 and G2 are same
 If G1 and G2 are isomorphic then number of
vertices of G1 and G2 are same
 If G1 and G2 are isomorphic then degree
sequence of G1 and G2 is same
 If G1 and G2 are isomorphic then both the
graph have a cycle of length k if present

6. Contrapositive

7.  If the number of edges of G1 and G2 are NOT
same then G1 and G2 are NOT isomorphic
 If the number of vertices of G1 and G2 are NOT
same then G1 and G2 are NOT isomorphic
 If the degree sequence of G1 and G2 is NOT
same then G1 and G2 are NOT isomorphic
 If G1 contain k cycle but G2 does NOT then G1
and G2 are NOT isomorphic

8. Connectivity

9. Connected Graph

10. An undirected graph is called connected if there
is a path between every pair of distinct vertices
of the graph. An undirected graph that is not
connected is called disconnected.

11. Connected Component

12. A connected component of a graph G is a
connected subgraph of G that is not a proper
subgraph of another connected subgraph of G.
That is, a connected component of a graph G is a
maximal connected subgraph of G. A graph G that
is not connected has two or more connected
components that are disjoint and have G as their
union.

13. Cut Vertex

14. A subset 𝑉⊥
of the vertex set V of G = (V; E) is a
vertex cut, or separating set, if G − 𝑉⊥
is
disconnected.

15. Vertex Connectivity

16. We define the vertex connectivity of a
noncomplete graph G, denoted by κ(G),as the
minimum number of vertices in a vertex cut. κ is
the lowercase Greek letter kappa