 
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,
kregular 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 kregular 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
