On the number of spanning trees of bicirculant graphs

Abstract

A bi-Cayley graph over a cyclic group $\mathbb{Z}_n$ is called a bicirculant graph. Let $\Gamma=BC(\mathbb{Z}_n; R,T,S)$ be a bicirculant graph with $R=R^{-1}\subseteq \mathbb{Z}_n\setminus \{0\}$ and $T=T^{-1}\subseteq \mathbb{Z}_n\setminus \{0\}$ and $S\subseteq \mathbb{Z}_n$. In this paper, using Chebyshev polynomials, we obtain a closed formula for the number of spanning trees of bicirculant graph $\Gamma$, investigate some arithmetic properties of the number of spanning trees of $\Gamma$, and find its asymptotic behaviour as $n$ tends infinity. In addition, we show that $F(x)=\sum_{n=1}^{\infty}\tau(\Gamma)x^n$ is a rational function with integer coefficients.