Junying Lu

Proofs of two conjectures on generalizations 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 $1\le k\le n$, $\sum_{i=1}^k\lambda_i(G)\le |E|+\binom{k+1}{2}$, which was recently confirmed by Kothari and Tudose. Before Brouwer's conjecture was proved, Lew (JCT-B, 2026) established a weaker form of Brouwer's Laplacian eigenvalue inequality and proposed two conjectures for upper bounds on the sum of the $k$ largest Laplacian eigenvalues, one in terms of the matching number and the other in terms of the vertex-cover number. Using Brouwer's Laplacian inequality, we prove both conjectures.