On a conjecture for the signless Laplacian eigenvalues

Abstract

Let $G$ be a simple graph with $n$ vertices and $e(G)$ edges, and $q_1(G)\ge q_2(G)\ge \cdots \ge q_n(G)\ge 0$ be the signless Laplacian eigenvalues of $G.$ Let $S_k^{+}(G)={\sum }_{i=1}^k{q}_i(G),$ where $k=1,2,\dots ,n.$ F. Ashraf et al. conjectured that $S_k^{+}(G)\le e(G)+\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ for $k=1,2,\dots ,n.$ In this paper, we give various upper bounds for $S_k^{+}(G),$ 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 $k\ne 3.$

13 pages, 5 figures