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arXiv · 2103.10580

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

Jiang-Chao Wan·Yi Wang·Ali Mohammadian
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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.

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Jiang-Chao Wan, Yi Wang, Ali Mohammadian. 2021-03-19. On the location of zeros of the Laplacian matching polynomials of graphs. https://arxiv.org/abs/2103.10580

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