A maximum matching based refinement of Brouwers conjecture

Let $G$ be a simple graph on $n$ vertices and $e(G)$ edges. Let ${\mu }_1\ge \cdots \ge {\mu }_{n-1}\ge {\mu }_n=0$ be the Laplacian eigenvalues of $G$. For $k=1,\dots ,n$, let $S_k(G)={\sum }_{i=1}^k{\mu }_i$. Brouwers conjecture asserts that for any $k\in \{1,\dots ,n\}$, $S_k(G)\le e(G)+\left(\genfrac{}{}{0px}{}{k+1}{2}\right)$. 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 $1\le k\le \lfloor g\mathrm{/}5\rfloor$, 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 $1\le k\le \lfloor g\mathrm{/}4\rfloor$. In this article, we further strengthen these results by proving that the Brouwers conjecture holds for $1\le k\le \lfloor \frac{m(G)}{2}\rfloor ,$ where $m(G)$ denotes the matching number of $G$. Since $m(G)\ge \lfloor g\mathrm{/}2\rfloor$, 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 $1\le k\le \lfloor \frac{n}{4}\rfloor$. 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 $h$, ${\mathcal{S}}_h(\bar{G})\le e(\bar{G})+\left(\genfrac{}{}{0px}{}{h+1}{2}\right)$ holds whenever ${\mathcal{S}}_h(G)\le e(G)+\left(\genfrac{}{}{0px}{}{h+1}{2}\right)$, where $\bar{G}$ denotes the complement of $G$.