发表机构
Trinity College, University of Cambridge（剑桥大学三一学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文改进了图最小特征值绝对值的递归下界，结合盈余估计完全解决了Räty-Sudakov-Tomon猜想，并给出了K_t-自由图盈余的指数下界，应用于MaxCut和Chowla余弦问题。
AI 中文摘要
Jin、Milojević、Tomon和Zhang建立了一个强大的递归估计，将最小特征值绝对值较小的图的正特征值联系起来。我们对该结果进行了改进，在谱图论、差异理论以及Chowla余弦问题中得到了改进的估计。Janzer、Tomon和Yip的最新结果使我们能够直接将最小特征值估计转化为相应的盈余估计。利用这些工具，我们完全解决了Räty、Sudakov和Tomon的一个猜想。我们证明，对于具有最小特征值λ_n和盈余sp(G)的n顶点图G，如果G与所有团的不相交并集ε-远离，则|λ_n|≥Ω_ε(n^{1/4})且sp(G)≥Ω_ε(n^{5/4})。此外，我们证明当G与所有团的不相交并集n^{-o(1)}-远离时，有|λ_n|≥n^{1/4 - o(1)}且sp(G)≥n^{5/4 - o(1)}。最后，我们证明具有m条边的K_t-自由图的盈余至少为m^{0.614 - o(1)}，当m趋于无穷时。
英文摘要
Jin, Milojević, Tomon and Zhang established a powerful recursive estimate relating the positive eigenvalues of a graph whose least eigenvalue is small in absolute value. We establish a refinement of this result, yielding improved estimates across spectral graph theory and discrepancy theory, and for Chowla's cosine problem. Recent results of Janzer, Tomon and Yip allow us to directly transfer least eigenvalue estimates to the corresponding surplus estimates. Using these tools, we fully resolve a conjecture of Räty, Sudakov and Tomon. We show that, for an $n$-vertex graph $G$ with least eigenvalue $λ_n$ and surplus $\operatorname{sp}(G)$, if $G$ is $ε$-far from all disjoint unions of cliques, then $|λ_n|\geq Ω_ε(n^{1/4})$ and $\operatorname{sp}(G)\geq Ω_ε(n^{5/4})$. Furthermore, we show that when $G$ is $n^{-o(1)}$-far from all disjoint unions of cliques, we have $|λ_n|\geq n^{1/4 - o(1)}$ and $\operatorname{sp}(G)\geq n^{5/4 - o(1)}$. Finally, we show that the surplus of a $K_t$-free graph with $m$ edges is at least $m^{0.614 - o(1)}$ as $m$ tends to infinity.
Comments41 pages
