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 $Q(G)=D+A$ as the signless Laplacian of $G$, 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 plus one edge. In this paper, we prove that inequality $q_1(G)+q_2(G)<=e(G)+3$ is tighter for the graph $S_n^{+}$ among all firefly graphs and also tighter to $S_n^{+}$ than to the graphs $K_k\vee \overline{K_{n-k}}$ recently presented by Ashraf, Omidi and Tayfeh-Rezaie. Also, we conjecture that the same inequality is tighter to $S_n^{+}$ than any other graph on $n$ vertices.