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 edges

To prove:

Prove that the property (has m edges) 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 edges.

We then need to show that G′ has m edges as well.

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 a one-to-one correspondence h between E ( G ) and E(G′). However, a one-to-one correspondence only exists between two sets that have the same number of elements