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
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; 
Subjects: 
Science & Engineering > 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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