GRAPH THEORY- I & II NOTES (1)

Example-2: Check whether the above two graphs are isomorphic
or not.

AB PQ

CD RS

Graph G1 Graph G2

i. Number of vertices=4 i. Number of vertices=4
ii. Number of edges=4 ii. Number of edges=4
iii. Degree sequence(A,B,C,D)= iii. Degree sequence
(P,Q,R,S)=(2,2,2,2)
(2,2,2,2)

iv. B2 S2
C2 P
A 2
2
R2

B2 R2
D Q

2 C2 2 S2

one-to-one correspondence: each vertex in graph G1 is

equivalent one vertex in Graph G2.

A  P, B  S, C  R, D  Q

v. Edge Preserving: each edge in graph G1 is equivalent one edge
in Graph G2.
A-B  P-S, A-C  P-R, C-D  R-Q, B-D  S-Q

vi. Adjacency matrices of both the graphs are
equal

A B CD P S RQ

A01 10 P 0 11 0

B10 01 S 1 00 1

C10 0 1 R 1 00 1

D01 1 0 Q 011 0

By satisfying all the above six properties mentioned above by
two graphs G1 and G2, Hence we can say that G1 and G2 are

Isomorphic graphs.

Example-3: Check whether the above two graphs are isomorphic or
not. P

BC ST

A D Q UR
F E

Graph G1 Graph G2

i. Number of vertices=6 i. Number of vertices=6
ii. Number of edges=9 ii. Number of edges=9
iii. Degree sequence of iii. Degree sequence of

(A,B,C,D,E,F)= (P,Q,R,S,T,U)=
(2,3,4,2,3,4) (2,2,2,4,4,4)

By observe the above two graphs, they have 6 vertices and 9

edges. But, the first graph has 2 vertices of degree 4 where as
second graph has 3 vertices of degree 4.

Therefore, there can not be any one-to-one correspondence
between the vertices and between the edges of the graphs which
preserves the adjacency of vertices. As such, the two graphs are
not isomorphic.

DIRAC’S THEOREM & ORE’S THEOREM

Dirac’s Theorem:

if G is a simple graph with n vertices(n>=3) such that
every vertex degree in G is atleast n/2 then graph G has a
Hamiltonian cycle, where n is the number of vertices.

Ore’s Theorem:

if G is a simple graph with n vertices(n>=3) such that
degree(u)+degree(v)>=n for every pair of non-adjacent
vertices u and v, where n is the number of vertices.

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