A Matching-Number Refinement of Brouwer's Laplacian Eigenvalue Inequality

Let $G=(V,E)$ be a finite simple graph with Laplacian eigenvalues
$\lambda_1(L(G))\ge\cdots\ge\lambda_{|V|}(L(G))$, and define
\[
\eps_k(G)=
\sum_{j=1}^{\min\{k,|V|\}}\lambda_j(L(G))-|E|.
\]
Let $\nu(G)$ be the matching number of $G$, and let $n(G)$ be the number of
non-isolated vertices of $G$. Lew proved that
\(\eps_k(G)\le k\nu(G)+\lfloor k/2\rfloor\), and conjectured that the
additive term can be removed in the non-endpoint range. We prove this
conjecture:
\[
\eps_k(G)\le k\nu(G)
\qquad
(1\le k\le n(G)-2).
\]
We also characterize all equality cases. Up to isolated vertices, equality
holds precisely for stars, for \(K_1\vee(K_k\cup\overline{K_{n-k-1}})\) with
\(k\) odd, and for \(K_n-E(K_{1,t})\) with \(n\) odd, \(k=n-2\), and
\(1\le t\le n-2\). We also analyze the endpoint range \(k\ge n(G)-1\), where
\(\eps_k(G)=|E|\), and determine the specific cases where the inequality \(\varepsilon_k(G)\le k\nu(G)\) fails or holds with equality.