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

A connected graph G is a cactus if any two of its cycles have at most one common vertex. Let l(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 l(n)(m) with n = 2m. In this paper, we characterize the case n >= 2m + 1. This confirms the conjecture of Li and Zhang (Li SC, Zhang MJ, On the signless Laplacian index of cacti with a given number of pendant vetices, Linear Algebra Appl. 2012; 436: 44004411). Further, we characterize the unique graph with the maximum signless Laplacian spectral radius among all cacti on n vertices.