Abstract
For a simple graph G(V, E) with n vertices, m edges, vertex set V(G) = {v(1), v(2),..., v(n)} and edge set E(G) = {e(1), e(2),..., e(m)}, the adjacency matrix A = (a(ij)) of G is a (0, 1)-square matrix of order n whose (i, j)-entry is equal to 1 if vi is adjacent to v(j) and equal to 0, otherwise. Let D(G) = diag(d(1), d(2),..., d(n)) be the diagonal matrix associated to G, where d(i) = deg(v(i)), for all i is an element of {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 = mu(n) <= mu(n-1) <= center dot center dot center dot <= mu(1) are the Laplacian eigenvalues of G, Brouwer conjectured that the sum of k largest Laplacian eigenvalues S-k(G) satisfies S-k(G) = Sigma(k)(i=1) mu(i) <= m + ((k+1)(2)) and this conjecture is still open. If q(1), q(2),..., q(n) are the signless Laplacian eigenvalues of G, for 1 <= k <= n, let S-k(+) (G) = Sigma(k)(i=1) q(i) be the sum of k largest signless Laplacian eigenvalues of G. Analogous to Brouwer's conjecture, Ashraf et al. conjectured that S-k(+) (G) <= m + ((k+1)(2)), 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 omega, the vertex covering number tau and the diameter of the graph G. Finally, we show that the conjecture holds for large families of graphs.