The number of rooted spanning forests of bicirculant graphs

A bi-Cayley graph over the 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 $R=-R\subseteq {\mathbb{Z}}_n\setminus \{0\}$ and $T=-T\subseteq {\mathbb{Z}}_n\setminus \{0\}$ and $S\subseteq {\mathbb{Z}}_n$. In this paper, using Chebyshev polynomials, we obtain a closed formula for the number of rooted spanning forests of $\mathrm{\Gamma }$. Moreover, we investigate some arithmetic properties of the number of rooted spanning forests of $\mathrm{\Gamma }$, and find its asymptotic behaviour as $n$ tends infinity.