Extremal graphs for the sum of the two largest signless Laplacian eigenvalues

Abstract

Let G be a simple graph on n vertices and e(G) edges. Consider the signless Laplacian, Q(G) = D + A, where A is the adjacency matrix and D is the diagonal matrix of the vertices degree of G. Let q_1(G) and q_2(G) be the first and the second largest eigenvalues of Q(G), respectively, and denote by S_n^+ the star graph with an additional edge. It is proved that inequality q_1(G)+q_2(G) \leq e(G)+3 is tighter for the graph S_n^+ among all firefly graphs and also tighter to S_n^+ than to the graphs K_k \vee K_{n−k} recently presented by Ashraf, Omidi and Tayfeh-Rezaie. Also, it is conjectured that S_n^+ minimizes f(G) = e(G) − q_1(G) − q_2(G) among all graphs G on n vertices.