Verify that the two graphs shown are isomorphic by exhibitin

Verify that the two graphs shown are isomorphic by exhibiting an isomorphism arrow diagram between their vertex sets. (b) The graph shown below is not isomorphic to the graphs given in part (a). Identify two isomorphism invariants that this graph does not share with the graphs in part (a).

Solution

If two graphs are isomorphic, they must have:
- the same number of vertices
- the same number of edges
- the same number of connected components
- the same number of loops.

-both graphs are connected or both graphs are not
connected,

-pairs of connected vertices must have the
corresponding pair of vertices connected

- the same degrees for corresponding vertices

Part-1)

To verify that the two graphs are isomorphic.

Proof-

if they are isomorphic then they must satisfy the above conditions,so lets check the conditions

Let us assume the two figure as A and B

vertices -----> fig(A)=6 fig(B)=6

Connected Components----> fig(A)=7 fig(B)=7

No. of Loops-----> fig(A)=3 fig(B)=3

Graphs arecoonected-----> fig(A)=Yes fig(B)=Yes

Edges -------> fig(A)=6 fig(B)=6

pairs of connected vertices must have the
corresponding pair of vertices connected---> fig(A) =Yes fig(B)=Yes

-Same Degrees for corresponding vertices--> Fig(A)=Yes fig(B)=Yes

so hence, They are isomorphic

Part-2

Assume the graph in part-2 as fig(C)

This graph does not have following things :