The number of rooted spanning forests of bicirculant graphs

TL;DR

This paper derives a closed-form formula for counting rooted spanning forests in bicirculant graphs using Chebyshev polynomials, explores their arithmetic properties, and analyzes their asymptotic behavior as the graph size grows.

Contribution

It introduces a novel formula for rooted spanning forests in bicirculant graphs and studies their arithmetic and asymptotic properties.

Findings

Derived a closed formula using Chebyshev polynomials.

Analyzed arithmetic properties of the forest count.

Investigated asymptotic behavior as n increases.

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.