On the structure of Laplace characteristic polynomial for circulant foliation

TL;DR

This paper analyzes the structure of the Laplace characteristic polynomial for a broad class of circulant foliation graphs, providing decomposition formulas and applications to spectral invariants like spanning trees.

Contribution

It introduces a novel decomposition of the Laplace characteristic polynomial for circulant foliation graphs, including generalized Petersen and torus graphs, into algebraic functions and polynomial products.

Findings

Characteristic polynomial decomposed into algebraic functions and Chebyshev roots

Representation of polynomial as a product involving integer polynomials

Derived formulas for spectral invariants such as spanning trees

Abstract

In this paper, we describe the structure of the Laplace characteristic polynomial $\chi_n(\lambda)$ 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 $\chi_n(\lambda)=p(\lambda)\,\chi_H(\lambda)a(n)^2,$ where $a(n)$ is a sequence of integer polynomials and $p(\lambda)$ 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.