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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 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
with
. In this paper, we characterize the case
. 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;436:4400–4411). Further, we characterize the unique graph with the maximum signless Laplacian spectral radius among all cacti on n vertices.
Notes
No potential conflict of interest was reported by the authors.
Additional information
Funding
L. You’s research is supported by the National Natural Science Foundation of China [grant number 11571123]; Guangdong Provincial Natural Science Foundation [grant number 2015A030313377], S. Li’s research is supported by the National Natural Science Foundation of China [grant number 11271149], Y. Shen’s research is supported by the Scientific Research Foundation of Graduate School of South China Normal University [grant number 2015lkxm19].