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

TL;DR

This paper proves a conjecture about the maximum signless Laplacian spectral radius in cacti graphs with a given matching number, extending previous results to larger graph sizes.

Contribution

It characterizes the extremal cacti graphs with maximum signless Laplacian spectral radius for all sizes, confirming a prior conjecture.

Findings

Confirmed the conjecture for n ≥ 2m+1.

Identified the unique extremal graph for maximum spectral radius.

Extended previous results from the case n=2m to larger n.

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.