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

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 C n (s 1 , s 2 , ... , s k) and C 2 n (s 1 , s 2 , ... , s k , n). Also, we prove that in both cases the number of spanning trees can be represented in the form τ (n) = p n a (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) = 2 k − ∑ j = 1 k (z s j + z − s j ). [ABSTRACT FROM AUTHOR]