The Number of Spanning Trees in Regular Graphs Summary: The Number of Spanning Trees in Regular Graphs Noga Alon* School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel ABSTRACT Let C(G) denote the number of spanning trees of a graph G . It is shown that there is a function ~ ( k )that tends to zero as k tends to infinity such that for every connected, k-regular simple graph G on n vertices C(G)= {k[l - u(G)]}",where 0 Iu(G)I~ ( k ) . 1. INTRODUCTION What is the minimum possible number of spanning trees of a k-regular connected simple graph on n vertices? This problem was suggested to me by P. Sarnak. His motivation came from number theory; there are many constructions of regular, connected graphs coming from number theoretic considerations (see, e.g., [8], [9]). In some of these constructions, one can obtain an expression for the class number of a certain function field in terms of the number of spanning trees of the corresponding graph. Therefore, the study of class numbers of function fields leads to a study of the number of spanning trees in regular, connected graphs. All graphs considered here are finite, undirected, and simple, i.e., have no loops and no multiple edges, unless otherwise specified. The complexity of a Collections: Mathematics