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Hoffman program of Laplacian matching polynomials of graphs

Part of: Graph theory

Published online by Cambridge University Press:  10 November 2025

Zhaoxi Li ,
Jianfeng Wang ,
Shi-Mei Ma  and
Francesco Belardo
Zhaoxi Li
Affiliation:
School of Mathematics and Statistics, Shandong University of Technology, Zibo, China (lzhaoxi@aliyun.com; jfwang@aliyun.com; shimeimapapers@163.com)
Jianfeng Wang
Affiliation:
School of Mathematics and Statistics, Shandong University of Technology, Zibo, China (lzhaoxi@aliyun.com; jfwang@aliyun.com; shimeimapapers@163.com)
Shi-Mei Ma
Affiliation:
School of Mathematics and Statistics, Shandong University of Technology, Zibo, China (lzhaoxi@aliyun.com; jfwang@aliyun.com; shimeimapapers@163.com)
Francesco Belardo
Affiliation:
Department of Mathematics and Applications, University ‘Federico II’, Naples, Italy (fbelardo@unina.it)
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Abstract

Let $\mathcal{G}$ be the class of all connected simple graphs. The Hoffman program of graphs with respect to a spectral invariant $\lambda(G)$ consists of determining all the limit points of the set $\{\lambda(G)\,\vert\, G\in\mathcal{G}\}$ and characterising all $G$’s such that $\lambda(G)$ does not exceed a fixed limit point. In this paper, we study the Hoffman program for Laplacian matching polynomials of graphs in regard to their largest Laplacian matching roots. Precisely, we determine all the limit points of the largest Laplacian matching roots of graphs less than $\tau = 2+\omega^{\frac{1}{2}}+\omega^{-\frac{1}{2}}(=4.38+)$, and then characterise the connected graphs with the largest Laplacian matching roots less than $2+\sqrt{5}$, where $\omega=\frac{1}{3}(\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}+1)$.

Keywords

matching polynomialLaplacian matching polynomiallimit pointlargest rootHoffman program

MSC classification

Primary: 05C31: Graph polynomials
Secondary: 05C70: Factorization, matching, partitioning, covering and packing

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 69 , Issue 2 , May 2026 , pp. 581 - 598
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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