On the sum of signless Laplacian spectra of graphs - Pirzada, S.; Ganie, H.A.; Alghamdi, A.M. (Carpathian Mathematical Publications, 2019)
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 v_i 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=(v_i), 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=μ_n≤μ_n-1≤…≤μ_1 are the Laplacian eigenvalues of G, Brouwer conjectured that the sum of k largest Laplacian eigenvalues S_k(G) satisfies S_k(G)=_i=1^kμ_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)=_i=1^kq_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 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.