A graph G is isomorphic to H if they are the same up to permuting the vertices. If you have a graph 1 -- 2 -- 3 (a path on 3 vertices, with 2 edges), it's struc

A graph G is isomorphic to H if they are the same up to permuting the vertices. If you have a graph 1 -- 2 -- 3 (a path on 3 vertices, with 2 edges), it's structurally the same as 3 -- 1 -- 2, only the labels of the vertices are different, and that doesn't matter for a graph theorist. So we say the graphs are isomorphic. I had a hard time understanding what it means when I was in undergraduate school, because it seems so trivial -- why can't I say that the graphs are simply the same, like 1 = 1 or {0} = {0}? You could say they are the same, but the interesting point is that it is not easy to decide whether they are the same if the two graphs G and H are big. Therefore, "graph isomorphism" is a computational problem in its own right.