A lower bound for the sum of the two largest signless Laplacian eigenvalues

Let G be a connected graph of order n≥3 and let Q(G)=D(G)+A(G) be the signless Laplacian of G, where A(G) is the adjacency matrix and D(G) is the diagonal matrix of the row-sums of A(G). Write q1(G) and q2(G) for the two largest eigenvalues of Q(G). In this paper, we obtain a lower bound to the sum of the two Q-largest eigenvalues, that is, q1(G)+q2(G)≥d1(G)+d2(G)+1 with equality if and only if G is the star Sn or the complete graph K3, where di is the i-largest degree of a vertex of G.