On complexity of cyclic coverings of graphs

TL;DR

This paper introduces a new analytic approach to compute the complexity of cyclic coverings of graphs, providing explicit formulas, asymptotic analysis, and properties of the generating function.

Contribution

It presents an explicit formula for counting the complexity of cyclic graph coverings using Chebyshev polynomials and analyzes its asymptotic behavior.

Findings

Derived an explicit formula for $ au(n)$ using Chebyshev polynomials

Established the asymptotic behavior of $ au(n)$ via Mahler measure

Proved the generating function $F(x)$ is rational with integer coefficients

Abstract

By complexity of a finite graph we mean the number of spanning trees in the graph. The aim of the present paper is to give a new approach for counting complexity $\tau(n)$ of cyclic $n$-fold coverings of a graph. We give an explicit analytic formula for $\tau(n)$ in terms of Chebyshev polynomials and find its asymptotic behavior as $n\to\infty$ through the Mahler measure of the associated voltage polynomial. We also prove that $F(x)=\sum\limits_{n=1}^\infty\tau(n)x^n$ is a rational function with integer coefficients.