On the Brouwer-type Conjecture for Signless Laplacian Eigenvalues of Graphs

Motivated by Brouwer's conjecture, Ashraf, Omidi and Tayfeh-Rezaie proposed the following Brouwer-type conjecture that for every graph $G$ on $n$ vertices with $m$ edges, the sum $S_k^+(G)$ of its $k$ largest signless Laplacian eigenvalues satisfies $S_k^+(G)\le m+\binom{k+1}{2}$ for $k=1, \ldots, n