On the number of spanning trees of bicirculant graphs

TL;DR

This paper derives a closed-form formula for counting spanning trees 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 closed-form expression for spanning trees in bicirculant graphs and studies their arithmetic and asymptotic properties.

Findings

Derived a closed formula for the number of spanning trees using Chebyshev polynomials.

Established that the generating function for spanning trees is a rational function with integer coefficients.

Analyzed the asymptotic growth of the number of spanning trees as the graph size increases.

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.