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 = deg (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 k q 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. [ABSTRACT FROM AUTHOR]