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

Abstract

For a simple graph G with n vertices and m edges, let $D(G)=$ diag$(d_1, d_2, \dots , d_n)$ be its diagonal matrix, where $d_i=\deg (v_i),$ for all $i=1,2,\dots ,n$ and A(G) be its adjacency matrix. The matrix $Q(G)=D(G)+A(G)$ is called the signless Laplacian matrix of G. If $q_1,q_2,\dots ,q_n$ are the signless Laplacian eigenvalues of Q(G) arranged in a non-increasing order, let $S^{+}_{k}(G)=\sum _{i=1}^{k}q_i$ 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 $k=1,2,\dots ,n$. 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.