发表机构
Sharif University of Technology; Center for Pure Mathematics, School of Mathematical Sciences, Anhui University（谢里夫理工大学; 安徽大学数学科学学院纯数学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究强化了连通图的平方能量相关结论，证明非圈2-连通图的正平方能量下界，还刻画了满足负平方能量下界的无三角2-连通图，明确了例外圈类型。
AI 中文摘要
设G是阶为n的图，s⁺(G)和s⁻(G)分别表示G的邻接特征值的正平方和与负平方和。最近，Liu、Tang和Zhang证明了Elphick、Farber、Goldberg与Wocjan的猜想：每个阶为n的连通图G都满足min{s⁺(G), s⁻(G)} ≥ n-1。对于正平方能量，我们强化了该结果，证明每个阶为n且非圈的2-连通图G都满足s⁺(G) ≥ n。而s⁻的形式类似断言不成立：完全图Kₙ满足s⁻(Kₙ)=n-1。我们在无三角图类中证明了一个自然的对应结论：每个无三角的非圈2-连通图G都满足min{s⁺(G),s⁻(G)}>n。更一般地，只需G的某个最大度顶点不属于任何三角形即可。结合圈的精确平方能量，这刻画了满足s⁻(G)≥n的无三角2-连通图；唯一的例外是k≥1的圈C_{4k+3}。
英文摘要
Let $G$ be a graph of order $n$, and let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of the positive and negative adjacency eigenvalues of $G$, respectively. Recently, Liu, Tang, and Zhang proved the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ of order $n$ satisfies $ \min\{s^+(G), s^-(G)\} \ge n-1. $ For positive square energy, we strengthen this result by showing that every $2$-connected graph $G$ of order $n$ which is not a cycle satisfies $s^+(G)\ge n$. The formally analogous assertion for $s^-$ is false: the complete graph $K_n$ satisfies $s^-(K_n)=n-1$. We prove a natural counterpart in the triangle-free class: every triangle-free $2$-connected noncycle $G$ satisfies $ \min\{s^+(G),s^-(G)\}>n. $ More generally, it is enough that some maximum-degree vertex of $G$ belongs to no triangle. Together with the exact square energies of cycles, this characterizes the triangle-free $2$-connected graphs for which $s^-(G)\ge n$; the only exceptions are the cycles $C_{4k+3}$ with $k\geq1$.
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