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