Two Conjectures on Extensions of Brouwer's Laplacian Conjecture

Let $G=(V,E)$ be a simple graph of order $n$ and let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ be the eigenvalues of its Laplacian matrix. Brouwer conjectured that for every $1\le k\le n$, $\sum_{i=1}^k\lambda_i(G)\le |E|+\binom{k+1}{2}$. Lew (JCTB, 2026) established a weaker form of Brouwer's Laplacian eigenvalue inequality. The full Brouwer conjecture was recently proved by Kothari and Tudose. Lew also proposed two conjectures for upper bounds on the sum of the largest Laplacian eigenvalues, one in terms of the matching number and one in terms of the vertex-cover number. Using Brouwer's Laplacian inequality, we prove both conjectures.