Some identities for enumerators of circulant graphs

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 graphComment: 17 pages, 3 tables Categories: CO Combinatorics (NT Number Theory) Math Subject Class: 05C30; 05A19; 11A4

Counting Unrooted Loopless Planar Maps

Abstract. We present a formula for the number of n-edge unrooted loopless planar maps considered up to orientation-preserving isomorphism. The only sum contained in this formula is over the divisors of n. Résumé. Nous présentons une formule pour le nombre de cartes planaires sans boucles avec n arêtes, à isomorphisme près préservant l’orientation. La seule somme contenue dans cette formule est prise parmi les diviseurs de n. 1

Address for correspondence:

Abstract. A planar map is a 2-cell embedding of a connected planar graph, loops and parallel edges allowed, on the sphere. A plane map is a planar map with a distinguished outside (“infinite”) face. An unrooted map is an equivalence class of maps under orientation-preserving homeomorphism, and a rooted map is a map with a distinguished oriented edge. Previously we obtained formulae for the number of unrooted planar n-edge maps of various classes, including all maps, non-separable maps, eulerian maps and loopless maps. In this article, using the same technique we obtain closed formulae for counting unrooted plane maps of all these classes and their duals. The corresponding formulae for rooted maps are known to be all sum-free; the formulae that we obtain for unrooted maps contain only a sum over the divisors of n. We count also unrooted two-vertex plane maps. 2000 Mathematics Subject Classification: 05C30 Key words: rooted planar map, unrooted plane map, quotient map, sum-free formula