The full Brouwer conjecture on equality cases

Let $G$ be a finite simple graph with $n$ vertices and $m$ edges. Let $\mu_1(G)\geq\mu_2(G)\geq\cdots\geq\mu_n(G)=0$ be its Laplacian eigenvalues, and define

$$S_k(G):=\sum_{i=1}^k\mu_i(G).$$

A threshold graph is a graph constructible from an isolated vertex by repeatedly adding either an isolated vertex or a dominating vertex, where a dominating vertex is adjacent to all preceding vertices. The full Brouwer conjecture. For every $k\in\{1,2,\dots,n-1\}$,

$$S_k(G)\leq m+\binom{k+1}{2},$$

with equality if and only if $G$ is a threshold graph with clique number $k+1$. This equality characterization is the central question addressed by the paper and is presented as a conjecture attributed to Li and Guo. The inequality itself is now a theorem by Kothari and Tudose, while the equality characterization is proved in the paper under consideration.