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.
Highlights
Spectral graph theory studies properties of graphs using the spectrum of related matrices
Let lmn denote the set of cacti with n vertices and matching number m
We characterize the unique graph with the maximum signless Laplacian spectral radius among all cacti on n vertices
Summary
Spectral graph theory (for example, [11, 12, 16] et al) studies properties of graphs using the spectrum of related matrices. Let G − v, G − uv denote the graph obtained from G by deleting vertex v ∈ V (G), or an edge uv ∈ E(G), respectively (this notation is naturally extended if more than one vertex or edge is deleted). Let lmn denote the set of cacti with n vertices and matching number m. This improves and confirms Conjecture 1.1 of Li and Zhang in [32]. We characterize the unique graph with the maximum signless Laplacian spectral radius among all cacti on n vertices. Let B1(Hsk) be the corresponding equitable quotient matrix of Q(Hsk)
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