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

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 $n\ge 2m+1$. 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.

arXiv comment20

References in corpus (2)

• Spectrum of a class of matrices and its applications
• Spectral Radius of Bipartite Graphs