On the Enumeration of Circulant Graphs of Prime Power and Square Free Orders

The aim of this work is twofold: to unify, systematize and extend the known results of the analytical (i.e. formula-wise) counting of directed and undirected circulant graphs with a prime or, more generally, square-free number of vertices; to develop a general combinatorial framework for the counting of nonisomorphic circulant graphs with pk vertices, p odd prime, k 2. The general problem of counting circulant graphs of order pk is decomposed into Cat(k) well-speci ed enumerative P olya type subproblems with respect to certain Abelian groups where Cat(k) denotes the k-th Catalan number. These subproblems are parametrized by the monotone underdiagonal walks on the plane integer (k + 1) (k + 1) lattice. The descriptions are given in terms of equalities and congruences between multipliers acting on sets of numbers in accordance with an isomorphism theorem for such circulant graphs. Tables contain new numerical results.