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

Gabriel Coutinho, Paula M. S. Fialho

TL;DR

This work introduces the multivariate sesquivalent polynomial $Φ_G(x,y,z)$, which interpolates between the matching polynomial and the Harary-Sachs expansion of the characteristic polynomial, and proves an explicit zero-free region for graphs of bounded maximum degree. The authors reinterpret $Φ_G$ as the partition function of a hard-core polymer gas, with polymers consisting of edges and cycle components and weights $oldsymbol{ω}(e)= y/x^2$, $oldsymbol{ω}(C_k)= z/x^k$, enabling the use of the Fernández–Procacci convergence criterion to guarantee nonvanishing in a specified region. Building on this, they develop a deterministic Barvinok–Patel–Regts interpolation scheme: reducing to a univariate interpolation via $F(t)= t^n Φ_G(x/t,y,z)$, proving zero-freeness on a disk, and computing the necessary Taylor coefficients through a BIGCP framework to approximate $Φ_G(x,y,z)$ and $ ext{log }Φ_G$ with controlled error. The paper further provides refinements under girth constraints and a specialization to $y=-1$, plus a concrete algorithm (Barvinok–Taylor on the canonical interpolation) with explicit running time dependent on the degree bound. Overall, the results advance both the understanding of zero-free regions for graph polynomials and practical deterministic schemes for evaluating them on bounded-degree graphs.

Abstract

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.

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

TL;DR

This work introduces the multivariate sesquivalent polynomial , which interpolates between the matching polynomial and the Harary-Sachs expansion of the characteristic polynomial, and proves an explicit zero-free region for graphs of bounded maximum degree. The authors reinterpret as the partition function of a hard-core polymer gas, with polymers consisting of edges and cycle components and weights , , enabling the use of the Fernández–Procacci convergence criterion to guarantee nonvanishing in a specified region. Building on this, they develop a deterministic Barvinok–Patel–Regts interpolation scheme: reducing to a univariate interpolation via , proving zero-freeness on a disk, and computing the necessary Taylor coefficients through a BIGCP framework to approximate and with controlled error. The paper further provides refinements under girth constraints and a specialization to , plus a concrete algorithm (Barvinok–Taylor on the canonical interpolation) with explicit running time dependent on the degree bound. Overall, the results advance both the understanding of zero-free regions for graph polynomials and practical deterministic schemes for evaluating them on bounded-degree graphs.

Abstract

We initiate the study of a multivariate graph polynomial 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 on bounded-degree graphs, obtained via the Fernández--Procacci convergence criterion for abstract polymer gases.
Paper Structure (15 sections, 9 theorems, 69 equations)

This paper contains 15 sections, 9 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Characteristic polynomial.
  3. Matching polynomial.
  4. Recurrences and reconstruction.
  5. Polymer-gas representation and proof of the zero-free region
  6. From sesquivalent subgraphs to a polymer partition function
  7. Polymers.
  8. Compatibility.
  9. Activities.
  10. Proof of the main theorem
  11. Ex.
  12. Barvinok interpolation and deterministic approximation
  13. Analytic reduction: zero-freeness and Taylor truncation
  14. Algorithmic step: computing the Taylor coefficients via Patel--Regts
  15. Algorithm (Barvinok--Taylor on the canonical interpolation).

Key Result

Theorem 1.3

Let $G$ be a finite graph with maximum degree $\Delta\ge 2$. Assume $x\in\mathbb{C}$ satisfies Then $\Phi_G(x,y,z)\neq 0$ whenever $(y,z)\in\mathbb{C}^2$ satisfy

Theorems & Definitions (22)

  • Definition 1.1: Sesquivalent (or elementary) subgraphs
  • Definition 1.2: Sesquivalent polynomial
  • Theorem 1.3: Zero-free region for $\Phi_G$
  • Theorem 2.1: Fernández--Procacci criterion, hard-core case FernandezProcacci07
  • proof : Proof of Theorem \ref{['thm:main']}
  • Remark 2.2: Girth refinement
  • Lemma 2.3
  • proof
  • Proposition 2.4: Linear admissible cycle parameter
  • proof
  • ...and 12 more