AI 中文总结
研究关于图的正、负邻接特征值平方和的猜想，通过引入新框架，将半正定矩阵哈达玛平方松弛到完全双非负锥，证明了\(n\)个顶点连通图满足\(\min\{s^+(G),s^-(G)\}\geq n - 1\)。
AI 中文摘要
设\(s^+(G)\)和\(s^-(G)\)分别表示图\(G\)的正、负邻接特征值的平方和。我们证明了埃尔菲克、法伯、戈德堡和沃恰恩的猜想，即每个\(n\)个顶点的连通图\(G\)满足\(\min\{s^+(G),s^-(G)\}\geq n - 1\)。证明引入了一个新的平方能量估计框架，其中编码这些谱量的半正定矩阵的哈达玛平方被松弛到完全双非负锥。
英文摘要
Let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of the positive and negative adjacency eigenvalues of a graph $G$, respectively. We prove the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies $$ \min\{s^+(G),s^-(G)\}\ge n-1. $$ The proof introduces a new framework for square-energy estimates, in which the Hadamard squares of positive semidefinite matrices that encode these spectral quantities are relaxed to the full doubly nonnegative cone.
Comments12 pages. Comments and suggestions are welcome
