  We establish analytically several new identities connecting enumerators of
different types of circulant graphs of prime, twice prime and prime-squared
orders. In particular, it is shown that the semi-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.
  Keywords: circulant graph; cycle index; cyclic group; nearly doubled primes;
Cunningham chain; self-complementary graph; tournament; mixed graph
