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.