Some Identities for Enumerators of Circulant Graphs

We establish analytically several new identities connecting enumerators of different types of circulant graphs mainly of prime, twice prime and prime-squared orders. In particular, it is shown that the half-sum of the number of undirected circulants and the number of undirected self-complementary circulants of prime order is equal to the number of directed self-complementary circulants of the same order. Several identities hold only for prime orders p such that (p + 1)/2 is also prime. Some conjectured generalizations and interpretations are discussed.