Characterizing the equality case in Brouwer's inequality for Laplacian eigenvalues

Yuhang Cui, Xiaodan Chen

Brouwer conjectured that the sum of the $k$ largest Laplacian eigenvalues of an $n$-vertex graph is less than or equal to the number of its edges plus $\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ for every $k…