I am working on a graph theory proof but facing problems regarding how to approach to prove it
Consider the following statement: Given a set of n graphs, {G1, G2, · · · , Gn}, some of the pair of graphs are isomorphic, yet some pair are not. There will be even number of these graphs which are isomorphic to an odd number of graphs. Prove or disprove the statement.
As for what I can figure out, if the even number of graphs are isomorphic to the odd ones, then vice versa will also be true, disproving the statement. But we are not sure if that would be the only thing.
Pardon if any mistakes made. Thanks in advance.
Consider the graph $G$ whose vertices are the graph $G_1, G_2, ..., G_n$ with two vertices adjacent if and only if they are isomorphic. The degree of a vertex is the number of graphs (other than itself) to which the vertex is isomorphic. By the handshaking lemma, there are an even number of vertices of odd degree.
The statement as given is false, since it overlooks the fact that each graph is isomorphic to itself.