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

Let $G$ be a graph of order $n\ge 3$ with sequence degree given as $d_1(G)\ge \mathrm{.}\mathrm{.}\mathrm{.}\ge d_n(G)$ and let ${\mu }_1(G),\mathrm{.}\mathrm{.}\mathrm{.},{\mu }_n(G)$ and $q_1(G), …, q_{n}(G)$betheLaplacianandsignlessLaplacianeigenvaluesof$G$ arranged in non increasing order, respectively. Here, we consider the Grone’s inequality [R. Grone, Eigenvalues and degree sequences of graphs, Lin. Multilin. Alg. 39 (1995) 133–136] $$\sum _{i=1}^k{\mu }_i(G)\ge \sum _{i=1}^k{d}_i(G)+1$$ and prove that for $k=2$, the equality holds if and only if $G$ is the star graph $S_n\mathrm{.}$ The signless Laplacian version of Grone’s inequality is known to be true when $k=1.$ In this paper, we prove that it is also true for $k=2,$ that is, $$q_1(G)+q_2(G)\ge d_1(G)+d_2(G)+1$$ with equality if and only if $G$ is the star $S_n$ or the complete graph $K_3\mathrm{.}$ When $k\ge 3$, we show a counterexample.