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

Abstract

Let $G=(V,E)$ be a finite simple graph with Laplacian eigenvalues $λ_1(L(G))\ge\cdots\geλ_{|V|}(L(G))$, and define \[ \eps_k(G)= \sum_{j=1}^{\min\{k,|V|\}}λ_j(L(G))-|E|. \] Let $ν(G)$ be the matching number of $G$, and let $n(G)$ be the number of non-isolated vertices of $G$. Lew proved that \(\eps_k(G)\le kν(G)+\lfloor k/2\rfloor\), and conjectured that the additive term can be removed in the non-endpoint range. We prove this conjecture: \[ \eps_k(G)\le kν(G) \qquad (1\le k\le n(G)-2). \] 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 \(\varepsilon_k(G)\le kν(G)\) fails or holds with equality.