The number of rooted spanning forests of bicirculant graphs

TL;DR

The paper addresses counting rooted spanning forests in bicirculant graphs by deriving closed formulas using Chebyshev polynomials and leveraging the circulant structure to diagonalize relevant matrices. It provides explicit expressions for f_Γ(2n) across the four bicirculant types and establishes arithmetic properties that express counts as squares times parity-dependent square-free factors, linked to the roots of associated Chebyshev transforms. Asymptotic growth is characterized via Mahler measures of the corresponding Laurent polynomials, yielding clear exponential rates for each bicirculant family. The work extends determinant-based spectral techniques from circulant to bicirculant graphs and connects combinatorial invariants to algebraic-number-theoretic concepts, with concrete examples including dihedral Cayley graphs to illustrate the results.

Abstract

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

Theorems & Definitions (11)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Theorem 3.1
• proof
• Theorem 4.1
• proof
• Theorem 5.1