Oliveira, Carla Silva and de Lima, Leonardo Silva and de Abreu, Nair Maria Maia and Kirkland, Steve (2010) Bounds on the Qspread of a graph. Linear Algebra and its Applications, 432 (9). pp. 23422351. ISSN 00243795
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Abstract
The spread s(M) of an n × n complex matrix M is s(M) = maxij _i − _j , where the maximum is taken over all pairs of eigenvalues of M, _i, 1 ≤ i ≤ n, [9] and [11]. Based on this concept, Gregory et al. [7] determined some bounds for the spread of the adjacency matrix A(G) of a simple graph G and made a conjecture regarding the graph on n vertices yielding the maximum value of the spread of the corresponding adjacency matrix. The signless Laplacian matrix of a graph G, Q(G) = D(G)+A(G), where D(G) is the diagonal matrix of degrees of G and A(G) is its adjacency matrix, has been recently studied, [4], [5]. The main goal of this paper is to determine some bounds on s(Q(G)). We prove that, for any graph on n ≥ 5 vertices, 2 ≤ s(Q(G)) ≤ 2n − 4, and we characterize the equality cases in both bounds. Further, we prove that for any connected graph G with n ≥ 5 vertices, s(Q(G)) < 2n − 4. We conjecture that, for n ≥ 5, sQ(G) ≤ √4n2 − 20n + 33 and that, in this case, the upper bound is attained if, and only if, G is a certain path complete graph.
Item Type:  Article 

Additional Information:  Preprint submitted to Elsevier 
Keywords:  spectrum; signless Laplacian matrix; spread; path complete graph; 
Academic Unit:  Faculty of Science and Engineering > Research Institutes > Hamilton Institute 
Item ID:  2187 
Depositing User:  Professor Steve Kirkland 
Date Deposited:  13 Oct 2010 15:34 
Journal or Publication Title:  Linear Algebra and its Applications 
Publisher:  Elsevier 
Refereed:  No 
URI: 
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