从平方能猜想到符号图：正平方能的紧界
From the Square-Energy Conjecture to Signed Graphs: Sharp Bounds for Positive Square Energy
AI总结：
该研究证明了连通图任意符号下正平方能的紧界，由此推出平方能猜想，还缩短了相关双非负矩阵不等式的证明。
AI中文摘要：
设Σ=(G,σ)为阶数n、边数m的连通符号图，s⁺(Σ)与s⁻(Σ)分别为其正、负邻接特征值的平方和。Elphick、Farber、Goldberg与Wocjan提出的平方能猜想指出，每个阶数n的连通图G均满足min{s⁺(G),s⁻(G)}≥n−1。Liu与Ning在题为《谱图理论中的未解决问题》的综述论文中，将该猜想列为该类问题的首位。我们证明，连通图G的任意符号σ均满足紧界s⁺(Σ)≤2m−n+1；对全正符号，此式给出s⁺(G)≤2m−n+1，对全负符号则给出s⁻(G)≤2m−n+1。由于s⁺(G)+s⁻(G)=2m，这两个特殊情形蕴含了平方能猜想；本定理的适用范围更广，因为同一界对G的所有符号均成立。将该定理应用于其否定−Σ，还可得到s⁺(Σ)≥n−1。两个界均为紧界。证明基于双非负矩阵不等式，我们还通过用固定凸组合替换其最终的情形区分，缩短了该不等式的证明过程。
英文摘要：
Let $Σ=(G,σ)$ be a connected signed graph of order $n$ and size $m$, and let $s^{+}(Σ)$ and $s^{-}(Σ)$ denote the sums of the squares of its positive and negative adjacency eigenvalues, respectively. The square-energy conjecture of Elphick, Farber, Goldberg, and Wocjan states that every connected graph $G$ of order $n$ satisfies \[ \min\{s^{+}(G),s^{-}(G)\}\ge n-1. \] Liu and Ning~\cite{LiuNing2023} published a wide-ranging paper entitled ``Unsolved Problems in spectral graph theory", and this conjectures were placed first in their list of such problems. We prove that every signature $σ$ of a connected graph $G$ satisfies the sharp bound \[ s^{+}(Σ)\le 2m-n+1. \] For the all-positive signing this gives $s^{+}(G)\le 2m-n+1$, whereas for the all-negative signing it gives $s^{-}(G)\le 2m-n+1$. Since $s^{+}(G)+s^{-}(G)=2m$, these two special cases imply the square-energy conjecture; the present theorem is stronger in scope because the same bound holds for every signing of $G$. Applying the theorem to the negation $-Σ$ also yields \[ s^{+}(Σ)\ge n-1. \] Both bounds are sharp. The proof is based on a doubly nonnegative matrix inequality. We also shorten the proof of that inequality by replacing its final case distinction with a fixed convex combination.
