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 ),