A note on the sum of the two largest signless Laplacian eigenvalues

For a simple graph G of order n with m edges, Ashraf et al. in 2013 conjectured that S-k* (G) <= m ((k+1)(2)) for k = 1, 2,..., n, where S-k*(G) = Sigma(k)(i=1) q(i) and q(1) >= q(2) >= center dot center dot center dot >= q(n) are the signless Laplacian eigenvalues of G. They gave a proof for the conjecture when k = 2, but applied an incorrect key lemma. In this note, we will give a corresponding counterexample to the key lemma. Moreover, we also prove that the conjecture is true for all connected triangle-free graphs when k = 2.