Enumeration of a Class of Rooted Planar Unicyclic Maps

Abstract

A unicyclic map is a rooted planar map such that there is only one cycle which is the boundary of the unique inner face (the inner face contains no trees) and the root-vertex is on the cycle. In this paper we investigate the number of unicyclic maps and present some formulae for such maps with up to three parameters: the number of edges and the valencies of the root-vertex and the root-face.