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

TL;DR

This paper introduces a new method to explicitly compute the number of spanning trees in circulant graphs, reveals their arithmetic structure, and derives asymptotic formulas using Mahler measures.

Contribution

It provides explicit formulas for spanning trees in circulant graphs and establishes their arithmetic and asymptotic properties.

Findings

Number of spanning trees expressed as  n a(n)^2 with integer sequence a(n)

Explicit formulas for  in various circulant graphs

Asymptotic behavior derived via Mahler measure

Abstract

In this paper, we develop a new method to produce explicit formulas for the number $\tau(n)$ of spanning trees in the undirected circulant graphs $C_{n}(s_1,s_2,\ldots,s_k)$ and $C_{2n}(s_1,s_2,\ldots,s_k,n).$ Also, we prove that in both cases the number of spanning trees can be represented in the form $\tau(n)=p \,n \,a(n)^2,$ 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 $\tau(n)$ through the Mahler measure of the associated Laurent polynomial $L(z)=2k-\sum\limits_{i=1}^k(z^{s_i}+z^{-s_i}).$