Counting circulant graphs of prime-power order by decomposing into orbit enumeration problems

Abstract

The set of circulant graphs with p k vertices ( k⩾1, p an odd prime) is decomposed into a collection of well-specified subsets. The number of these subsets is equal to the kth Catalan number, and they are in one-to-one correspondence with the monotone underdiagonal walks on the plane integer (k+1)×(k+1) lattice. The counting of non-isomorphic circulant graphs in each of the subsets is presented as an orbit enumeration problem of Pólya type with respect to a certain Abelian group of multipliers. The descriptions are given in terms of equalities and congruences between multipliers in accordance with an isomorphism theorem for such circulant graphs. In this way, explicit uniform counting formulae have been obtained for various types of circulant graphs with p 2 vertices.