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

We initiate the study of a multivariate graph polynomial ${\mathrm{\Phi }}_G(x,y,z)$ 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 ${\mathrm{\Phi }}_G$ on bounded-degree graphs, obtained via the Fernández–Procacci convergence criterion for abstract polymer gases.