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

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.