Zero-freeness of a multivariate monomer-dimer-cycle polynomial on bounded-degree graphs

TL;DR

This paper introduces a multivariate graph polynomial that bridges matchings and cycle structures, establishing a zero-free region for bounded-degree graphs using polymer gas convergence criteria.

Contribution

It provides the first explicit zero-free region for the polynomial on bounded-degree graphs, connecting combinatorial polynomials with analytic properties.

Findings

Established a zero-free region for the polynomial on bounded-degree graphs.

Connected the polynomial's properties to the Fernandez-Procacci convergence criterion.

Bridged classical counting polynomials for matchings and cycle structures.

Abstract

We initiate the study of a multivariate graph polynomial $\Phi_G(x,y,z)$ that interpolates between classical counting polynomials for matchings and for cycle structures arising in the Harary--Sachs expansion of the characteristic polynomial. We focus on analytic properties and computational consequences. Our main contribution is an explicit, degree-uniform zero-free region for $\Phi_G$ on bounded-degree graphs, obtained via the Fern\'andez--Procacci convergence criterion for abstract polymer gases.