On the number of spanning trees of bicirculant graphs

TL;DR

The paper develops closed-form expressions for the number of spanning trees $\tau(\Gamma)$ of bicirculant graphs $\Gamma=BC(\mathbb{Z}_n;R,T,S)$ by leveraging Chebyshev polynomials and associated transforms. It expresses $\tau(\Gamma)$ in terms of leading coefficients of polynomials $P_1,P_2,P_3,P_4$ and Chebyshev evaluations, and reveals arithmetic structure in which $\tau(2n)$ factors into squares up to square-free parts. The authors establish asymptotic growth rates via Mahler measures of the corresponding Laurent polynomials, and prove the generating function $F(x)=\sum_{n\ge1}\tau(\Gamma)x^n$ is rational with integer coefficients and symmetry. They illustrate the results with concrete examples of four bicirculant families, deriving explicit $\tau(\Gamma)$ formulas, asymptotics, and generating functions. The findings advance explicit enumeration of spanning trees in highly symmetric graphs and connect spectral, algebraic, and analytic techniques for combinatorial graph invariants.

Abstract

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

Theorems & Definitions (22)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• proof
• Lemma 2.8
• proof