On the location of zeros of the Laplacian matching polynomials of graphs

TL;DR

This paper studies the zeros of the Laplacian matching polynomial of graphs, revealing their location properties, interlacing behavior, and bounds on the largest zero, with implications for graph structure analysis.

Contribution

It introduces new results on the zeros of the Laplacian matching polynomial, including their characterization, interlacing properties, and bounds, advancing understanding of this graph polynomial.

Findings

Zero at zero iff the graph is a tree

Number of positive zeros ≥ length of longest path

Zeros of the polynomial and its edge-deletion version interlace

Abstract

The Laplacian matching polynomial of a graph $G$, denoted by $\mathscr{L\hspace{-0.7mm}M}(G,x)$, is a new graph polynomial whose all roots are nonnegative real numbers. In this paper, we investigate the location of zeros of the Laplacian matching polynomials. Let $G$ be a connected graph. We show that $0$ is a root of $\mathscr{L\hspace{-0.7mm}M}(G, x)$ if and only if $G$ is a tree. We prove that the number of distinct positive zeros of $\mathscr{L\hspace{-0.7mm}M}(G,x)$ is at least equal to the length of the longest path in $G$. It is also established that the zeros of $\mathscr{L\hspace{-0.7mm}M}(G,x)$ and $\mathscr{L\hspace{-0.7mm}M}(G-e,x)$ interlace for each edge $e$ of $G$. Using the path-tree of $G$, we present a linear algebraic approach to investigate the largest zero of $\mathscr{L\hspace{-0.7mm}M}(G,x)$ and particularly to give tight upper and lower bounds on it.