The number of rooted spanning forests of bicirculant graphs

Abstract

A bi-Cayley graph over the cyclic group $(\mathbb{Z}_n, +)$ is called a bicirculant graph. Let $\Gamma=BC(\mathbb{Z}_n; R,T,S)$ be a bicirculant graph with $R=-R\subseteq \mathbb{Z}_n\setminus \{0\}$ 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 $\Gamma$. Moreover, we investigate some arithmetic properties of the number of rooted spanning forests of $\Gamma$, and find its asymptotic behaviour as $n$ tends infinity.