On the Ky Fan norm of the signless Laplacian matrix of a graph

For a simple graph G with n vertices and m edges, let D(G)= diag (d_1, d_2, … , d_n) be its diagonal matrix, where d_i= (v_i), for all i=1,2,… ,n and A ( G ) be its adjacency matrix. The matrix Q(G)=D(G)+A(G) is called the signless Laplacian matrix of G . If q_1,q_2,… ,q_n are the signless Laplacian eigenvalues of Q ( G ) arranged in a non-increasing order, let S^+_k(G)=∑ _i=1^kq_i be the sum of the k largest signless Laplacian eigenvalues of G . As the signless Laplacian matrix Q ( G ) is a positive semi-definite real symmetric matrix, so the spectral invariant S^+_k(G) actually represents the Ky Fan k -norm of the matrix Q ( G ). Ashraf et al. (Linear Algebra Appl 438:4539–4546, 2013) conjectured that , for all k=1,2,… ,n . In this paper, we obtain upper bounds to S^+_k(G) for some infinite families of graphs. Those structural results and tools are applied to show that the conjecture holds for many classes of graphs, and in particular for graphs with a given clique number.