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Closed-form expansions for the universal edge elimination polynomial

Klaus Dohmen

TL;DR

The paper studies the universal edge elimination polynomial ξ(G,x,y,z), a generalization of several graph polynomials, and derives closed-form expansions for ξ on path and cycle graphs. By converting the recursive definitions into a linear recurrence and solving via the characteristic equation, it treats all discriminant cases to yield explicit formulas and generating functions. The results also yield corollaries for the bivariate matching polynomial, the bivariate chromatic polynomial, and the covered components polynomial, enabling direct computation and insights into these polynomials. Overall, the work provides exact, usable expressions for key graph families and clarifies relationships among several classical polynomials.

Abstract

We establish closed-form expansions for the universal edge elimination polynomial of paths and cycles and their generating functions. This includes closed-form expansions for the bivariate matching polynomial, the bivariate chromatic polynomial, and the covered components polynomial.

Closed-form expansions for the universal edge elimination polynomial

TL;DR

The paper studies the universal edge elimination polynomial ξ(G,x,y,z), a generalization of several graph polynomials, and derives closed-form expansions for ξ on path and cycle graphs. By converting the recursive definitions into a linear recurrence and solving via the characteristic equation, it treats all discriminant cases to yield explicit formulas and generating functions. The results also yield corollaries for the bivariate matching polynomial, the bivariate chromatic polynomial, and the covered components polynomial, enabling direct computation and insights into these polynomials. Overall, the work provides exact, usable expressions for key graph families and clarifies relationships among several classical polynomials.

Abstract

We establish closed-form expansions for the universal edge elimination polynomial of paths and cycles and their generating functions. This includes closed-form expansions for the bivariate matching polynomial, the bivariate chromatic polynomial, and the covered components polynomial.

Paper Structure

This paper contains 2 sections, 3 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Closed-form expansions for paths and cycles

Key Result

Theorem 2.1

Let $n\in\mathbb{N}$, and $x,y,z\in \mathbb{R}$. If $z > - \left( \frac{x+y}{2} \right)^2$, then where If $z < - \left( \frac{x+y}{2} \right)^2$, then where If $z = - \left( \frac{x+y}{2} \right)^2$, then

Theorems & Definitions (8)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Corollary 2.5
  • proof