Self-complementary graph

A self-complementary graph is a graph which is isomorphic to its complement. The simplest self-complementary graphs are the 4-vertex path graph and the 5-vertex cycle graph.

Self-complementary graphs are interesting in their relation to the graph isomorphism problem: the problems of checking whether two self-complementary graphs are isomorphic and of checking whether a given graph is self-complementary are polynomial-time equivalent to the general graph isomorphism problem.

An n-vertex self-complementary graph has exactly half number of edges of the complete graph, i.e., $n(n-1)/4$ and diameter 2 or 3.