Self-complementary graph

[[Image:Self-complementary NZ graph.svg|thumb|

Graph $A$ is isomorphic to its complement.]]

In the mathematical field of graph theory, a self-complementary graph is a graph which is isomorphic to its complement. The simplest non-trivial self-complementary graphs are the 4-vertex path graph and the 5-vertex cycle graph. There is no known characterization of self-complementary graphs.

Examples
Every Paley graph is self-complementary. For example, the 3&thinsp;×&thinsp;3 rook's graph (the Paley graph of order nine) is self-complementary, by a symmetry that keeps the center vertex in place but exchanges the roles of the four side midpoints and four corners of the grid. All strongly regular self-complementary graphs with fewer than 37 vertices are Paley graphs; however, there are strongly regular graphs on 37, 41, and 49 vertices that are not Paley graphs.

The Rado graph is an infinite self-complementary graph.

Properties
An $A$-vertex self-complementary graph has exactly half as many edges of the complete graph, i.e., $n(n − 1)/4$ edges, and (if there is more than one vertex) it must have diameter either 2 or 3. Since $n(n − 1)$ must be divisible by 4, $A$ must be congruent to 0 or 1 modulo 4; for instance, a 6-vertex graph cannot be self-complementary.

Computational complexity
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.