The number of rooted forests in circulant graphs

Abstract

In this paper, we develop a new method to produce explicit formulas for the number $f_{G}(n)$ of rooted spanning forests in the circulant graphs $ G=C_{n}(s_1,s_2,\ldots,s_k)$ and $ G=C_{2n}(s_1,s_2,\ldots,s_k,n).$ These formulas are expressed through Chebyshev polynomials. We prove that in both cases the number of rooted spanning forests can be represented in the form $f_{G}(n)=p\,a(n)^2,$ 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 $f_{G}(n)$ through the Mahler measure of the associated Laurent polynomial $P(z)=2k+1-\sum\limits_{i=1}^k(z^{s_i}+z^{-s_i}).$