An *n*-vertex self-complementary graph has exactly half number of 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.[1] Since *n*(*n* −1) must be divisible by 4, *n* must be congruent to 0 or 1 mod 4; for instance, a 6-vertex graph cannot be self-complementary.

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