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 vertices of degree k

To prove:

Prove that the property (has m vertices of degree 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 vertices of degree k (with k a positive integer). We then need to show that G′ has m vertices of degree 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

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 the m distinct vertices of degree k in G be  v 1 , v 2 ,..., v m . since  v i  has degree k(for i=1,2,...,m),

there exist p distinct loops  e 1 , e 2 ,..., e p  at  v i  and k−2p edges  e p+1 , e p+2 ,... e p+( k−2p )  with  v i  as endpoint.

However, h( e 1 ),h( e 2 ),...,h( e p ) are then p loops at g( v i ) and h( e p+1 ),h( e p+2 ),