Bounds on the Q-spread of a graph

The spread s(M) of an n x n complex matrix M is s(M) = max(ij)vertical bar lambda(i) - lambda(j)vertical bar, where the maximum is taken over all pairs of eigenvalues of M, lambda(i), 1 <= i <= n [E. Jiang, X. Zhan, Lower bounds for the spread of a hermitian matrix, Linear Algebra Appl. 256 (1997) 153-163; J. Kaarlo Merikoski, R. Kumar, Characterizations and lower bounds for the spread of a normal matrix, Linear Algebra Appl. 364 (2003) 13-31]. Based on this concept, Gregory et al. [D.A. Gregory, D. Hershkowitz, S.J. Kirkland, The spread of the spectrum of a graph, Linear Algebra Appl. 332-334 (2001) 23-35] 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 [D. Cvetkovie, Signless Laplacians and line graphs, Bulletin T. CXXXI de l' Academie serbe des sciences et des arts (2005) Classe des Sciences mathematiques et naturelles Sciences mathernatiques 30 (2005) 85-92; D. Cvetkovic, P. Rowlinson, S. Simic, Signless Laplacian of finite graphs, Linear Algebra Appl. 423 (2007)155-171]. The main goal of this paper is to determine some bounds on the spread of Q (G), which we denote by 5 (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, s(Q) (G) <= root 4n(2) - 20n + 33 and that, in this case, the upper bound is attained if, and only if. G is a certain path complete graph. (C) 2009 Elsevier Inc. All rights reserved.