Spanning Trees and Function Classes

Abstract

If $G=K_n$ is the complete graph, the classical Prüffer correspondence gives a natural bijection between all spanning trees of $G$ (i.e., all Cayley trees) and all functions from a set of $n-2$ elements to a set of $n$ elements. If $G$ is a complete multipartite graph, then such bijections have been studied by Eğecioğlu and Remmel. In this paper, we define a class of directed graphs, called filtered digraphs, and describe a natural class of bijections between oriented spanning forests of these digraphs and associated classes of functions. We derive multivariate generating functions for the oriented spanning forests which arise in this context, and we link basic properties of these spanning forests to properties of the functions to which they correspond. This approach yields a number of new results for directed graphs. Moreover, in the undirected case, various specializations of our multivariate generating function not only include various known results but also give a number of new results.