Subdivision method in the Laplacian matching polynomial

TL;DR

This paper explores the subdivision method's role in understanding the Laplacian matching polynomial, revealing majorization relations, dualities, and new combinatorial interpretations of its coefficients.

Contribution

It establishes a majorization relation between the Laplacian matching polynomial's zeros and the degree sequence, and offers new combinatorial insights into its coefficients.

Findings

Zero sequence of Laplacian matching polynomial majorizes degree sequence

Dual relation between Laplacian matching polynomial and signless Laplacian characteristic polynomial

New combinatorial interpretations for polynomial coefficients

Abstract

As a bridge connecting the matching polynomial and the Laplacian matching polynomial of graphs, the subdivision method is expected to be useful for investigating the Laplacian matching polynomial. In this paper, we study applications of the method from three aspects. We prove that the zero sequence of the Laplacian matching polynomial of a graph majorizes its degree sequence, establishing a dual relation between the Laplacian matching polynomial and the characteristic polynomial of the signless Laplacian matrix of graphs. In addition, from different viewpoints, we give a new combinatorial interpretations for the coefficients of the Laplacian matching polynomial.