Authors: Belardo, Francesco
Li Marzi, Enzo
Simić, Slobodan 
Wang, Jianfeng
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Graphs whose signless Laplacian spectral radius does not exceed the Hoffman limit value
Journal: Linear Algebra and Its Applications
Volume: 435
Issue: 11
First page: 2913
Last page: 2920
Issue Date: 1-Dec-2011
Rank: M22
ISSN: 0024-3795
DOI: 10.1016/j.laa.2011.05.006
For a graph matrix M, the Hoffman limit value H(M) is the limit (if it exists) of the largest eigenvalue (or, M-index, for short) of M(Hn), where the graph Hn is obtained by attaching a pendant edge to the cycle Cn-1 of length n-1. In spectral graph theory, M is usually either the adjacency matrix A or the Laplacian matrix L or the signless Laplacian matrix Q. The exact values of H(A) and H(L) were first determined by Hoffman and Guo, respectively. Since Hn is bipartite for odd n, we have H(Q)=H(L). All graphs whose A-index is not greater than H(A) were completely described in the literature. In the present paper, we determine all graphs whose Q-index does not exceed H(Q). The results obtained are determinant to describe all graphs whose L-index is not greater then H(L). This is done precisely in Wang et al. (in press) [21].
Keywords: Hoffman limit value | Index | Limit point of the eigenvalues | Signless Laplacian matrix | Spectral radius
Publisher: Elsevier
Project: PRIN 2008 (Italy) “Disegni Combinatorici, Grafi e loro Applicazioni”
Serbian Ministry of Science, Grant no. 144015
Graph theory and mathematical programming with applications in chemistry and computer science 
National Science Foundation of China (No. 10961023)
NSFQH (SRIPQHNU) (No. 2011-ZR-616)

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