A note on the sum of the largest signless Laplacian eigenvalues

Let G be a graph with n vertices and e(G) edges. The signless Laplacian of G, denoted by Q(G), is given by Q(G)=D(G)+A(G), where D(G) and A(G) are the diagonal matrix of its vertex degree and A(G) is the adjacency matrix. Let q1(G),…,qn(G) be the eigenvalues of Q(G) in non-increasing order and let Tk(G)=∑i=1kqi(G) be the sum of the k largest signless Laplacian eigenvalues of G. In this paper, we obtain an upper bound to Tk(H), when H is the P3-join graph isomorphic to P3[(n−k−1)K1,Kk−1,K2] for 3≤k≤n−2. Also, we conjecture that Tk(G) is bounded above by Tk(H) for any G with n vertices.