What does it mean for two graphs to be the same? Let G and H be graphs. We say that G is isomorphic to H provided that there is a bijection f:V(G) -> V(H) so that for all a, b, in V(G) there is an edge connecting a and b (in G) if and only if there is an edge connecting f(a) and f(b) (in H). The function f is called an isomorphism of G to H.

We can think of f as renaming the vertices of G with the names of the vertices of H in a way that preserves adjacency. Less formally, isomorphic graphs have the same drawing (except for the names of the vertices).

Do the following:

(a) Prove that isomorphic graphs have the same number of vertices.

(b) Prove that if f:V(G) -> V(H) is an isomorphism of graphs G and H and if v is an element of V(G), then the degree of v in G equals the degree of f(v) in H.

(c) Prove that isomorphic graphs have the same number of edges.

(d) Give an example of two non-isomorphic graphs that have the same number of vertices and the same number of edges.