Can the number of edges determine isomorphism in graphs?

[AlbertEinstein]

Hi all,

If I have to prove that the graph G and its complement G' are isomorphic, then is it enough to prove that both G and G' will have the same number of edges. Intuitively its clear to me, but how do I prove this. If there's a counterexample, please post.

Thanks in advance.

 

[disregardthat]

Hi all,

If I have to prove that the graph G and its complement G' are isomorphic, then is it enough to prove that both G and G' will have the same number of edges. Intuitively its clear to me, but how do I prove this. If there's a counterexample, please post.

Thanks in advance.

That is certainly not enough. Consider the graphs of four vertices and three edges. They are not isomorphic to each of their complements.
To prove an isomorphism you will need to define a function f : G --> G' such that an edge between $$v_1$$ and $$v_2$$ implies that there is an edge between $$f(v_1)$$ and $$f(v_2)$$.

 

[AlbertEinstein]

Oh yeah, I got the point.

Thanks for the help.