发表机构
IIT Bhubaneswar（印度理工学院布巴内斯瓦尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过匹配数改进Brouwer猜想的成立范围，证明当k不超过匹配数一半时猜想成立，并应用于完美匹配图，同时给出基于补图的等价验证条件。
AI 中文摘要
设$G$为具有$n$个顶点和$e(G)$条边的简单图。设$\mu_1\geq \cdots \geq \mu_{n-1}\geq \mu_n=0$为$G$的拉普拉斯特征值。对于$k=1, \ldots, n$，令$S_k(G)=\sum_{i=1}^{k}\mu_i$。Brouwer猜想断言：对于任意$k\in\{1,\ldots, n\}$，$S_k(G)\leq e(G)+\binom{k+1}{2}$。在文献[图的拉普拉斯特征值之和的上界，{\em Discrete Appl. Math.}, 170:95--103, (2014)]中，Rocha和Trevisan证明了该猜想对于$1 \leq k \leq \lfloor g/5 \rfloor$成立，其中$g$表示$G$的围长。这一关于$k$的界后来被Chen在[图的拉普拉斯特征值之和的Brouwer猜想的改进结果，{\em Linear Algebra Appl.}, 557:327--338, (2018)]中改进，他证明了该猜想对于$1 \leq k \leq \lfloor g/4 \rfloor$成立。在本文中，我们通过证明Brouwer猜想对于$1\leq k\leq\left\lfloor \frac{m(G)}{2}\right\rfloor$成立来进一步加强这些结果，其中$m(G)$表示$G$的匹配数。由于$m(G)\geq \lfloor g/2\rfloor$，该情形构成了对前述结果的真正改进。作为应用，我们证明：若$G$是阶为$n$且具有完美匹配的图，则Brouwer猜想对于$1\leq k\leq \left\lfloor \frac{n}{4}\right\rfloor$成立。最后，我们通过证明$G$满足Brouwer猜想当且仅当对于固定的正整数$h$，只要$\mathcal{S}_h(G)\leq e(G)+\binom{h+1}{2}$成立，就有$\mathcal{S}_h(\overline{G})\leq e(\overline{G})+\binom{h+1}{2}$成立（其中$\overline{G}$表示$G$的补图），为验证Brouwer猜想提供了新视角。
英文摘要
Let $G$ be a simple graph on $n$ vertices and $e(G)$ edges. Let $μ_1\geq \cdots \geq μ_{n-1}\geq μ_n=0$ be the Laplacian eigenvalues of $G$. For $k=1, \ldots, n$, let $S_k(G)=\sum_{i=1}^{k}μ_i$. Brouwers conjecture asserts that for any $k\in\{1,\ldots, n\}$, $S_k(G)\leq e(G)+\binom{k+1}{2}$. In [Bounding the sum of the largest Laplacian eigenvalues of graphs, {\em Discrete Appl. Math.}, 170:95--103, (2014)], Rocha and Trevisan showed that the conjecture holds true for $1 \leq k \leq \lfloor g/5 \rfloor$, where $g$ denotes the girth of $G$. This bound on $k$ was later improved by Chen in [Improved results on Brouwers conjecture for sum of the Laplacian eigenvalues of a graph, {\em Linear Algebra Appl.}, 557:327--338, (2018)], who established that the conjecture holds for $1 \leq k \leq \lfloor g/4 \rfloor$. In this article, we further strengthen these results by proving that the Brouwers conjecture holds for $1\leq k\leq\left\lfloor \frac{m(G)}{2}\right\rfloor,$ where $m(G)$ denotes the matching number of $G$. Since $m(G)\geq \lfloor g/2\rfloor$, the case constitutes a genuine improvement over the aforementioned results. As an application, we show that if $G$ is a graph of order $n$ and with a perfect matching, then the Brouwers conjecture holds for $1\leq k\leq \left\lfloor \frac{n}{4}\right\rfloor$. Finally, we provide a new perspective on verifying Brouwers conjecture by proving that $G$ satisfies the Brouwers conjecture if and only if for a fixed positive integer $h$, $\mathcal{S}_h(\overline{G})\leq e(\overline{G})+\binom{h+1}{2}$ holds whenever $\mathcal{S}_h(G)\leq e(G)+\binom{h+1}{2}$, where $\overline{G}$ denotes the complement of $G$.
