A maximum matching based refinement of Brouwers conjecture

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 [Bou