Bounds on the Q-spread of a graph

The spread s(M) of an n×n complex matrix M is s(M)=maxij|λi-λj|, where the maximum is taken over all pairs of eigenvalues of M, λi,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. Cvetković, Signless Laplacians and line graphs, Bulletin T. CXXXI de l’ Académie serbe des sciences et des arts (2005) Classe des Sciences mathématiques et naturelles Sciences mathématiques 30 (2005) 85–92; D. Cvetković, P. Rowlinson, S. Simić, 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 sQ(G). We prove that, for any graph on n⩾5 vertices, 2⩽sQ(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, sQ(G)<2n-4. We conjecture that, for n⩾5, and that, in this case, the upper bound is attained if, and only if, G is a certain path complete graph.