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 n vertices

To prove:

Prove that the property ( n vertices )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 n vertices.

We then need to show that G′ has n vertices as well.

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

Since G and G′ are isomorphic, there exists a one-to-one correspondence g between V ( G ) and V(G′). However, a one-to-one correspondence only exists between two sets that have the same number of elements