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

Let $G=(V,E)$ be a finite simple graph with Laplacian eigenvalues ${\lambda }_1(L(G))\ge \cdots \ge {\lambda }_{\mathrm{\mid }V\mathrm{\mid }}(L(G))$, and define $${\text{eps}}_k(G)=\sum _{j=1}^{\min \{k,\mathrm{\mid }V\mathrm{\mid }\}}{\lambda }_j(L(G))-\mathrm{\mid }E\mathrm{\mid }\mathrm{.}$$ 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 ${\text{eps}}_k(G)\le k\nu (G)+\lfloor k\mathrm{/}2\rfloor$, and conjectured that the additive term can be removed in the non-endpoint range. We prove this conjecture: $${\text{eps}}_k(G)\le k\nu (G)(1\le k\le n(G)-2)\mathrm{.}$$ 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 ${\text{eps}}_k(G)=\mathrm{\mid }E\mathrm{\mid }$, and determine the specific cases where the inequality ${\epsilon }_k(G)\le k\nu (G)$ fails or holds with equality.