发表机构
University of Bonn（波恩大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了Spielman和Teng的30年猜想，得到了亏格为$g$的图和$K_h$-无小图的拉普拉斯等特征值的最优或近最优界，推广了相关论证并利用了Lovász局部引理。
AI 中文摘要
本文解决了Spielman和Teng提出的一个已有30年的猜想，该猜想关于谱划分方法在可嵌入亏格为$g$的可定向曲面的图上的性能。具体而言，对于这类有$n$个顶点、最大度为$\text{Delta}$的图$G$，我们证明其拉普拉斯矩阵的第二小特征值满足$\lambda_2(L_G)\lesssim\Delta\frac{g}{n}$。我们还为$K_h$-无小图图获得了改进的特征值界，即$\lambda_2(L_G)\lesssim\Delta\frac{h^2(\log h)^2}{n}$。事实上，我们的结果直接为重新加权特征值（包括更高阶的重新加权特征值）证明了强得多的结论。因此，我们不仅得到了拉普拉斯特征值的界，还得到了归一化拉普拉斯特征值和Steklov特征值的界。我们关于亏格为$g$的图的结果对所有这些类型的特征值都是最优的，而我们关于$K_h$-无小图图的结果在$\log(h)$因子范围内是最优的。我们用于亏格为$g$的图的技术，将整个类别的归一化特征值的有界度界，自举为相同类别的无有界度限制的重新加权特征值的界；而我们用于$K_h$-无小图图的技术推广了Korhonen和Lokshtanov的论证，利用了Lovász局部引理。
英文摘要
In this paper, we resolve a 30-year-old conjecture of Spielman and Teng concerning the performance of the spectral partitioning method on graphs embeddable on an orientable surface of genus $g\ge 1$. In particular, for such a graph $G$ with $n$ vertices and maximum degree $Δ$, we show that the second-smallest eigenvalue of its Laplacian matrix satisfies $λ_2(L_G)\lesssimΔ\frac g n$. We also obtain an improved eigenvalue bound for $K_h$-minor-free graphs of $λ_2(L_G)\lesssimΔ\frac{h^2(\log h)^2}n$. In fact, our results directly prove much stronger results for reweighted eigenvalues, including higher reweighted eigenvalues. As a consequence, we obtain bounds not just on Laplacian eigenvalues, but also on normalized Laplacian eigenvalues and Steklov eigenvalues. Our results for genus-$g$ graphs are optimal for all of these kinds of eigenvalues, while our results for $K_h$-minor-free graphs are optimal up to $\log(h)$ factors. Our techniques for genus-$g$ graphs bootstrap bounded-degree bounds of normalized eigenvalues for entire classes to bounds for reweighted eigenvalues for the same classes without the bounded-degree limitation, while our techniques for $K_h$-minor-free graphs generalize an argument of Korhonen and Lokshtanov, making use of the Lovász local lemma.
