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.

Is connected

To prove:

Prove that the property (is connected) 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 be connected.

We then need to show that G′ is connected 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 v and w be two distinct vertices in G'. Since g is a one-to-one correspondence,there exist distinct vertices a and b such that g(a)=v and g(b)=w.Since G is connected, there exists a walk ae1v1e2v2