Isomorphic graphs.
Theorem
Let G1 and G2 be simple graphs. The following statements are equivalent: a) G1 and G2 are isomorphic b) There is a one-to-one function and on f, from the set of vertices of G1 to the set of vertices of G2 that satisfies the following condition: The vertices of v and w are adjacent in G1 if and only if the vertices f (v) and f (w) are adjacent in G2.
Demonstration
Let’s define a function g, from the edges of G1 to the edges of G2 using the rule
g ((v, w)) = (f (v), f (w))
Since G1 and G2 are simple graphs, neither have parallel edges, so the notation (v ‘, w’) unambiguously designates an edge. Note that the range of g is a subset of the edges of G2, since if (v, w) is an edge at G1, v and w are adjacent, which implies that f (v) and f (w) are adjacent; that is to say that (f (v), f (w)) is an edge in G2
