On the structure of Laplace characteristic polynomial for circulant foliation
Young Soo Kwon, Alexander Mednykh, Ilya Mednykh
TL;DR
The paper characterizes the Laplace characteristic polynomial $\chi_n(\lambda)$ for circulant foliations $H_n$ by deriving a Chebyshev-based product form $\chi_n(\lambda)=\eta(\lambda)^n\prod_{p=1}^s\bigl(2T_n(w_p)-2\bigr)$ where $w_p$ are roots of $Q_\lambda(w)=0$, and by proving a square-factor structure $\chi_n(\lambda)=\chi_H(\lambda)\,a_n(\lambda)^2$ (odd $n$) or $\chi_n(\lambda)=Q_\lambda(-1)\chi_H(\lambda)\,a_n(\lambda)^2$ (even $n$). This enables analytic formulas for spectral graph invariants, including the number of spanning trees $\tau(H_n)$ and rooted spanning forests $F(H_n)$, expressed through $|\eta(0)|^n$, $|\eta(-1)|^n$, and products of terms $|2T_n(w_p)-2|$. The approach unifies a broad class of graphs (e.g., generalized Petersen graphs, $I$-, $Y$-, $H$-graphs, discrete tori, Cartesian products) and provides concrete, computable expressions for key combinatorial invariants via determinant polynomials $P_\lambda(z)$ and their Chebyshev transforms $Q_\lambda(w)$.
Abstract
In this paper, we describe the structure of the Laplace characteristic polynomial $χ_n(λ)$ for the infinite family of graphs $H_n=H_n(G_1,\,G_2,\ldots,G_m)$ obtained as a circulant foliation over a graph $H$ on $m$ vertices with fibers $G_1,\,G_2,\ldots,G_m.$ Each fiber $G_i=C_n(s_{i,1},\,s_{i,2},\ldots,s_{i,k_i})$ of this foliation is the circulant graph on $n$ vertices with jumps $s_{i,1},\,s_{i,2},\ldots,s_{i,k_i}.$ This family includes the family of generalized Petersen graphs, $I$-graphs, sandwiches of circulant graphs, discrete torus graphs and others. We show that the characteristic polynomial for such graphs can be decomposed into a finite product of algebraic functions evaluated at the roots of a linear combination of Chebyshev polynomials. Also, we prove that the characteristic polynomial can be represented in the form $χ_n(λ)=p(λ)\,χ_H(λ)a(n)^2,$ where $a(n)$ is a sequence of integer polynomials and $p(λ)$ is a prescribed integer polynomial. Moreover, we use the obtained results to produce analytic formulas for spectral graph invariants, such as the number of spanning trees and the number of spanning rooted forests.