More accurate definition of isomorphism should also take into account colored, directed or weighted graphs. In this way, there should also be a one-to-one correspondence between graph's arcs and their properties with the property that the vertex and its adjacent arcs and its image in other graphs are in the same relationships with adjacent arcs. (I tried to generate the phrase on the fly, sorry if my phrasing is not so accurate).
See also:
http://en.wikipedia.org/wiki/Graph_theory[
^] and references.
—SA