On a conjecture for the sum of Laplacian eigenvalues

Let $G$ be a simple graph with $n$ vertices and $e\left(G\right)$ edges. Brouwer et al. conjectured that the sum of the $k$ largest Laplacian eigenvalues of $G$ is at most $e\left(G\right)+\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ , where $1\le k\le n$ . In this paper, this conjecture is proved to be true for the following cases: connected graphs with sufficiently large $k$ , unicyclic graphs, bicyclic graphs and tricyclic graphs with some restrictions, forests, etc. Moreover, we show that if $G$ is a tree with a specified property, then the sum of the $k$ largest Laplacian eigenvalues of $G$ is at most $e\left(G\right)+2k‐2$ , where $1\le k\le n$ .