On a conjecture for the signless Laplacian eigenvalues

Let G be a simple graph with n   vertices and e(G)e(G) edges, and q1(G)⩾q2(G)⩾⋯⩾qn(G)⩾0q1(G)⩾q2(G)⩾⋯⩾qn(G)⩾0 be the signless Laplacian eigenvalues of G  . Let Sk+(G)=∑i=1kqi(G), where k=1,2,…,nk=1,2,…,n. F. Ashraf et al. conjectured that Sk+(G)⩽e(G)+(k+12) for k=1,2,…,nk=1,2,…,n. In this paper, we give various upper bounds for Sk+(G), and prove that this conjecture is true for the following cases: connected graph with sufficiently large k, unicyclic graphs and bicyclic graphs for all k  , and tricyclic graphs when k≠3k≠3. Finally, we discuss whether the upper bound given in this conjecture is tight or not for c-cyclic graphs and propose some problems for future research.