Prove that each of the properties in 21-29 is an invariant for graph isomorphism. Assume that n, m, and k are all nonnegative integers.

Has m simple circuits of length k

To prove:

Prove that the property (has m simple circuits of length k ) is an invariant for graph isomorphism.

Given information:

Assume that n, m and k are all nonnegative integers.

Proof:

Let G and G′ be isomorphic graphs and let the graph G contain m simple circuits of length k (with k a positive integer). We then need to show that G′ has m simple circuits of length k as well.

Let V ( G ) be the set of vertices of G and let V(G′) be the set of vertices of G′.

Let E(G) be the set of edges of G and let E ( G’ ) be the set of edges of G′.

Since G and G′ are isomorphic, there exists two one-to-one correspondences

Thus G′ then contains m simple circuits of length k as well and thus having m simple circuits of length k is an invariant for graph isomorphism.

g:V( G )→V( G' ) and h:E( G )→E( G' ) such that 

v is an endpoint of e⇔g( v ) is an endpoint of h( e ).

Let a simple circuit of length k in G be  v 1 e 1 v 2 e 2 .... v k e k v 1  with  v i ∈V( G ) and  e i ∈E( G )

 for i=1,2,...,k. This then implies that the walk contains at least one edge and  e 1 , e 2 ,..., e k

contains no repeated edges and  v 1 , v 2 ,..., v k  contains no repeated vertices.

g( v 1 )h( e 1 )g( v 2 )h( e 2 )....g( v k )h( e k )g( v 1 ) is then a walk of length k in G', because v is 

an endpoint of e⇔g( v ) is an endpoint of h( e )