On the sum of signless Laplacian spectra of graphs

Abstract

For a simple graph G(V, E) with n vertices, m edges, vertex set V(G) = {v1, v2, …, vn} and edge set E(G) = {e1, e2, …, em}, the adjacency matrix A = (ai j) of G is a (0, 1)-square matrix of order n whose (i, j)-entry is equal to 1 if vi is adjacent to vj and equal to 0, otherwise. Let D(G) = diag(d1, d2, …, dn) be the diagonal matrix associated to G, where di = deg(vi), for all i ∈ {1, 2, …, n}. The matrices L(G) = D(G) − A(G) and Q(G) = D(G) + A(G) are respectively called the Laplacian and the signless Laplacian matrices and their spectra (eigenvalues) are respectively called the Laplacian spectrum (L-spectrum) and the signless Laplacian spectrum (Q-spectrum) of the graph G. If 0 = µn ≤ µn−1 ≤ · · · ≤ µ1 are the Laplacian eigenvalues of G, Brouwer conjectured that the sum of k largest Laplacian eigenvalues Sk (G) satisfies (Formula Presented) and this conjecture is still open. If q1, q2, …, qn are the signless Laplacian eigenvalues of G, for 1 ≤ k ≤ n, let (Formula Presented) be the sum of k largest signless Laplacian eigenvalues of G. Analogous to Brouwer’s conjecture, Ashraf et al. conjectured that (Formula Presented), for all 1 ≤ k ≤ n. This conjecture has been verified in affirmative for some classes of graphs. We obtain the upper bounds for S+k (G) in terms of the clique number ω, the vertex covering number τ and the diameter of the graph G. Finally, we show that the conjecture holds for large families of graphs.