Turán定理正平方能量强化中的等号情形
A Vertex-Localized Positive Square-Energy Strengthening of Turán's Theorem
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中文总结 AI 辅助
该研究刻画了Liu等人证明的Turán定理正平方能量强化不等式取等号的图族，得出r≥2时等号成立的充要条件为r整除n且G是完全正则r部图的结论。
中文摘要 AI 辅助
设G为阶数为n的图，其特征值为λ₁(G)≥…≥λₙ(G)，令s₊(G)=Σ_{λᵢ(G)>0}λᵢ(G)²。近期Liu、Tang和Zhang证明了Turán定理的正平方能量强化：√s₊(G)≤(1−1/r)n，其中r=ω(G)是G的团数。本文刻画了该不等式取等号的图族，确切证明：当r≥2时，等号成立当且仅当r整除n，且G是完全正则r部图K_{n/r,…,n/r}。
英文摘要
Let $G$ be a graph of order $n$ with the adjacency eigenvalues $λ_1(G) \geq \dots \geq λ_n(G) $. Let $c(v)$ denote the maximum order of a clique containing vertex $v$. We prove the vertex-localized positive square-energy inequality \[ \sqrt{s_+(G)} \leq \sum_{v\in V}\left(1-\frac1{c(v)}\right), \] where \[ s_+(G)=\sum_{λ_i(G)>0}λ_i(G)^2. \] We also characterize equality. Apart from edgeless graphs, equality holds precisely for graphs obtained from a complete regular multipartite graph by adding an arbitrary number of isolated vertices. This settles a conjecture of Kannan, Kumar and Pragada.
