AI 中文总结
该研究确定了Elphick等人关于图的正、负邻接特征值平方和最小值下界猜想的所有等号情形，通过结合相关引理与矩阵分析完成证明，明确了等号对应的图类。
AI 中文摘要
对于图$G$，令$s^+(G)$和$s^-(G)$分别表示其正邻接特征值平方和与负邻接特征值平方和。我们确定了Elphick、Farber、Goldberg和Wocjan猜想中的所有等号情形，该猜想指出，每个含$n$个顶点的连通图$G$满足$\text{min}\{s^+(G),s^-(G)\}\ge n-1$。具体而言，$s^+$取等当且仅当$G$是树，而$s^-$取等当且仅当$G$是树或完全图。证明过程结合了无割点情形下的$P_3$-删除引理，以及对 underlying 双非负矩阵不等式的详细等号分析。每个块被强制为完全图，且极小反例论证给出了折叠矩阵$M^c$的精确秩1分解。所得的非边消失条件，连同$AX=XA$，排除了桥与非平凡块之间的相互作用。
英文摘要
For a graph $G$, let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of its positive and negative adjacency eigenvalues. We determine all equality cases in the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies \[ \min \{s^+(G),s^-(G)\}\ge n-1. \] Namely, equality for $s^+$ holds exactly for trees, whereas equality for $s^-$ holds exactly for trees and complete graphs. The proof combines the $P_3$-removal lemma in the no-cut-vertex case with a detailed equality analysis of the underlying doubly nonnegative matrix inequality. Every block is forced to be complete, and a minimal-counterexample argument gives an exact rank-one decomposition of the folded matrix $M^c$. The resulting non-edge vanishings, together with $AX=XA$, rule out an interface between a bridge and a nontrivial block.
Comments19 pages
