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arXiv 2005.04502math.COmath.GR
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无有界特征基的Cayley图

Cayley graphs without a bounded eigenbasis

Ashwin Sah, Mehtaab Sawhney, Yufei Zhao

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18 pages
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摘要：

是否每个$n$ -顶点Cayley图都有一个其坐标都是$O(1/\sqrt{n})$的正交本征基？虽然对阿贝尔群的回答是肯定的，但我们证明了一般情况下它是否定的。另一方面，我们证明了每一个$n$ -顶点Cayley图（更一般地说，顶点传递图）有一个规范正交基，其坐标都是$O(\sqrt{\log n / n})$，并且这个界几乎是最好的可能。我们的调查是出于一个问题的Assaf Naor，谁证明了随机阿贝尔凯莱图是小集扩张，扩展了经典的结果Alon-Roichman。他的证明依赖于阿贝尔凯莱图的有界本征基的存在，我们现在知道这对一般群不成立。另一方面，我们绕过这个障碍，并扩展Naor的结果nonabelian组。

英文摘要：

Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon--Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor's result to nonabelian groups.

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