标题： 布劳威尔猜想的部分证明 显示英文标题

标题： A partial proof of the Brouwer's conjecture

摘要： 设$G$是一个具有$n$个顶点和$m$条边的简单图，令$k$是一个自然数，使得$k\leq n.$，Brouwer 猜测$G$的前$k$个最大拉普拉斯特征值之和不超过$m+{k+1 \choose 2}.$。在本文中，我们证明该猜想对于简单的$(m,n)$-图成立，其中$n\leq m\leq \frac{\sqrt{3}-1}{4}(n-1)n$且$k\in \left[ \sqrt[3]{\frac{8m^{2}}{n-1}+4mn+n^{2}}, n\right].$。此外，我们证明该猜想对于所有简单的$(m,n)$-图成立，其中$k (\leq n)$是来自区间$\left[\sqrt{2n-2m+2\sqrt{2m^{2}+mn(n-1)}},1+\frac{8m^{2}}{n^{2}(n-1)}+\frac{4m}{n}\right].$的自然数。 显示英文摘要

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