On the number of spanning trees of bicirculant graphs

A bi-Cayley graph over a cyclic group ${\mathbb{Z}}_n$ is called a bicirculant graph. Let $\mathrm{\Gamma }=BC({\mathbb{Z}}_n;R,T,S)$ 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 $\mathrm{\Gamma }$, investigate some arithmetic properties of the number of spanning trees of $\mathrm{\Gamma }$, and find its asymptotic behaviour as $n$ tends infinity. In addition, we show that $F(x)={\sum }_{n=1}^{\mathrm{\infty }}\tau (\mathrm{\Gamma })x^n$ is a rational function with integer coefficients.