The Equality Cases For the Laplacian Conjecture of Brouwer

Abstract

The Laplacian conjecture of Brouwer asserts that for any graph \(G\) of order n with \(m\) edges, the sum of the \(k\) largest Laplacian eigenvalues satisfies \(s_k(G) \le m + \binom{k+1}{2}\) for $k=1, \ldots, n$. Later, Li and Guo in 2022 further proposed the full Brouwer's Laplacian spectrum conjecture. Recently, Kothari and Tudose in 2026 proved the Brouwer's conjecture. Motivated by their perfect proof and methods, we proved that for a simple graph of order $n$ with $m$ edges and $1\le k\le n-1$, \(s_k(G) = m + \binom{k+1}{2}\) if and only if $G$ is a threshold graph with clique number \(k+1\), which confirms the full Brouwer conjecture proposed by Li and Guo.