第k个特征值的极值图
Extremal graphs for the $k$-th eigenvalue
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中文总结 AI 辅助
本文研究Hong问题中λₖ(G)的最优上界，刻画Sivashankar定理的等号情形，还给出λ₃和λ₄极值图的显式组合描述。
中文摘要 AI 辅助
对于n阶简单图G，设λ₁(G)≥…≥λₙ(G)为其邻接特征值。Hong问题求λₖ(G)的最优上界，Sivashankar近期定理给出对任意k≥3，λₖ(G)≤[(k-2)√(k+1)+2]/[2k(k-1)]·n -1，尖锐例子来自极大实等角紧框架。本文刻画等号情形，还得到λ₃和λ₄极值图的显式组合描述。
英文摘要
For a simple graph $G$ of order $n$, let $λ_1(G)\ge \cdots \ge λ_n(G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $λ_k(G)$. A recent theorem of Sivashankar gives, for every $k\ge3$, \[ λ_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1, \] with sharp examples arising from maximal real equiangular tight frames. In this paper, we characterize the equality case. We also obtain an explicit combinatorial description of the extremal graphs for $λ_3$ and $λ_4$.
