# Why is the graph of 4 nodes and 2 edges not self-complementary?

I am having some trouble seeing why a graph of 4 nodes and 2 edges is not self-complementary such that $$G$$ is isomorphic to $$\overline{G}$$ (G complement) (please see the attachment below). I know that the number of edges in a self-complementary graph must be $$\frac{1}{4}n(n-1)$$, (half of that of a $$K_n$$ graph) and so for a graph of 4 nodes, the number of edges must be 3. However, why is the graph attached not self-complementary?

The number of edges, nodes, the degree sequence, length of cycles, etc., are the same, and could I not map $$G$$ and $$G$$ complement like: $$f(d) = b$$; $$f(b) = d$$; $$f(a) = a$$; $$f(c) = c$$?

I must be missing something about isomorphism here - if someone could point it out, that'd be great - thanks so much!

• You’re missing the fact that $\overline G$ also has the edges $ac$ and $bd$. – Brian M. Scott Feb 2 '15 at 0:12
• I don't think you are missing something about isomorphism, I think you are missing something about complement. Your second graph is not the complement of your first. – David Feb 2 '15 at 0:13
• Oh wow, that was a terrible oversight - thanks so much! – Christine Feb 2 '15 at 0:14