The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic

Abstract

In this paper, we develop a new method to produce explicit formulas for the number τ(n) of spanning trees in the undirected circulant graphs Cn(s1,s2,…,sk) and C2n(s1,s2,…,sk,n). Also, we prove that in both cases the number of spanning trees can be represented in the form τ(n)=pna(n)2, where a(n) is an integer sequence and p is a prescribed natural number depending on the parity of n. Finally, we find an asymptotic formula for τ(n) through the Mahler measure of the associated Laurent polynomial L(z)=2k−∑j=1k(zsj+z−sj).