Proofs of two conjectures on generalizations of Brouwer's Laplacian conjecture

Junying Lu, Jiabao Yang

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 $1\le k\le n$, $\s…