Closed-form expansions for the universal edge elimination polynomial

TL;DR

This paper derives explicit formulas for the universal edge elimination polynomial and related graph polynomials for paths and cycles, facilitating easier computation and analysis of these graph invariants.

Contribution

It provides the first closed-form expansions for the universal edge elimination polynomial and several related bivariate graph polynomials for paths and cycles.

Findings

Closed-form expansions for the universal edge elimination polynomial.

Explicit formulas for bivariate matching and chromatic polynomials.

Generating functions for these polynomials are also derived.

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.