A partial proof of the Brouwer's conjecture

Let $G$ be a simple graph with $n$ vertices and $m$ edges and let $k$ be a natural number such that $k\le n\mathrm{.}$ Brouwer conjectured that the sum of the $k$ largest Laplacian eigenvalues of $G$ is at most $m+\left(\genfrac{}{}{0px}{}{k+1}{2}\right)\mathrm{.}$ In this paper we prove that this conjecture is true for simple $(m,n)$-graphs where $n\le m\le \frac{\sqrt{3}-1}{4}(n-1)n$ and $k\in [\sqrt[3]{\frac{8m^2}{n-1}+4mn+n^2},n]\mathrm{.}$ Moreover, we prove that the conjecture is true for all simple $(m,n)$-graphs where $k(\le n)$ is a natural number from the interval $[\sqrt{2n-2m+2\sqrt{2m^2+mn(n-1)}},1+\frac{8m^2}{n^2(n-1)}+\frac{4m}{n}]\mathrm{.}$