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arXiv · 1511.06902

On a conjecture for the signless Laplacian spectral radius of cacti with given matching number

Yun Shen·Lihua You·Minjie Zhang·Shuchao Li
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Abstract

A connected graph $G$ is a cactus if any two of its cycles have at most one common vertex. Let $\ell_n^m$ be the set of cacti on $n$ vertices with matching number $m.$ S.C. Li and M.J. Zhang determined the unique graph with the maximum signless Laplacian spectral radius among all cacti in $\ell_n^m$ with $n=2m$. In this paper, we characterize the case $n\geq 2m+1$. This confirms the conjecture of Li and Zhang(S.C. Li, M.J. Zhang, On the signless Laplacian index of cacti with a given number of pendant vetices, Linear Algebra Appl. 436, 2012, 4400--4411). Further, we characterize the unique graph with the maximum signless Laplacian spectral radius among all cacti on $n$ vertices.

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Yun Shen, Lihua You, Minjie Zhang, Shuchao Li. 2015-11-21. On a conjecture for the signless Laplacian spectral radius of cacti with given matching number. https://arxiv.org/abs/1511.06902

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