AI 中文总结
该研究针对图的第 $k$ 大邻接特征值，通过归约为正交投影极值问题，结合修正后的投影常数上界，得到紧的特征值上界并在特定偶数维实现严格改进。
AI 中文摘要
对于整数 $k\ge2$，设 $\lambda_k(G)$ 表示图 $G$ 的第 $k$ 大邻接特征值。对于每个有 $n$ 个顶点的图 $G$ 以及每个满足 $2 \leq k \leq n$ 的整数 $k$，我们证明：\\[ \lambda_k(G) \le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\\,n-1. \\] 该界对于 $k\in\{2,3,4,8,24\}$ 是紧的。我们通过将图特征值问题归约为正交投影的极值问题，再应用 Deręgowska 和 Lewandowska 给出的绝对投影常数 $\gamma(r)$ 的一般上界，得到了上述结果。我们还通过修正 König 和 Tomczak-Jaegermann 的 Gegenbauer 多项式论证，给出了他们的界的另一种证明。由此得到的松弛恒等式在每个满足 $r+2$ 不是完全平方数的偶数维 $r\ge4$ 中实现了严格改进。
英文摘要
For an integer $k\ge2$, let $λ_k(G)$ denote the $k$th largest adjacency eigenvalue of a graph $G$. For every graph $G$ on $n$ vertices and every $2 \leq k \leq n$, we prove \[ λ_k(G) \le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1. \] Our bound is tight for $k\in\{2,3,4,8,24\}$. We obtain it by reducing the graph-eigenvalue problem to an extremal problem for orthogonal projections and then applying the general upper bound on the absolute projection constant $γ(r)$ due to Deręgowska and Lewandowska. We also give an alternative proof of their bound by repairing the Gegenbauer-polynomial argument of König and Tomczak-Jaegermann. The resulting slack identity yields a strict improvement in every even dimension $r\ge4$ for which $r+2$ is not a perfect square.
CommentsThis paper combines and supersedes the manuscripts arXiv:2603.21181, arXiv:2603.28738, and arXiv:2603.29280, which will not be published separately, and includes additional results and improvements
