图的等周图谱及其逼近
On The Isoperimetric Spectrum of Graphs and Its Approximations
摘要：
本文研究了（有限有向）图的高阶等周数。在这方面，我们专注于$n$ th平均等周常数的有向图作为最小的平均传出归一化流从一组给定的$n$不相交的子集的顶点集的图形。我们表明，第二平均等周常数在这个一般的设置，符合（平均版本）的经典Cheeger常数的图，而对于其余的频谱，我们表明，有一个根本的区别之间的$n$第等周常数和通过采取最小的所有$n$-分区。在这个方向上，我们表明，我们的定义是正确的，在这个意义上说，它满足Federerer-Fleming型定理，我们还定义了一个超几何图的概念，并提出了例子，作为一个图的平均等周常数达到分区在所有级别。此外，考虑到图上的等周问题的${\bf NP}$-完备性，我们致力于逼近问题，我们证明了一般谱不等式，从而产生一个一般的Cheeger型不等式。另一方面，我们也考虑了一些算法方面的问题，我们显示连接到图的正交表示和以下J.~马利克和J。Shi（$2000$）提出的方法研究了与著名的$k$-均值算法和归一化割方法的密切关系。
英文摘要：
In this paper we consider higher isoperimetric numbers of a (finite directed) graph. In this regard we focus on the $n$th mean isoperimetric constant of a directed graph as the minimum of the mean outgoing normalized flows from a given set of $n$ disjoint subsets of the vertex set of the graph. We show that the second mean isoperimetric constant in this general setting, coincides with (the mean version of) the classical Cheeger constant of the graph, while for the rest of the spectrum we show that there is a fundamental difference between the $n$th isoperimetric constant and the number obtained by taking the minimum over all $n$-partitions. In this direction, we show that our definition is the correct one in the sense that it satisfies a Federer-Fleming-type theorem, and we also define and present examples for the concept of a supergeometric graph as a graph whose mean isoperimetric constants are attained on partitions at all levels. Moreover, considering the ${\bf NP}$-completeness of the isoperimetric problem on graphs, we address ourselves to the approximation problem where we prove general spectral inequalities that give rise to a general Cheeger-type inequality as well. On the other hand, we also consider some algorithmic aspects of the problem where we show connections to orthogonal representations of graphs and following J.~Malik and J.~Shi ($2000$) we study the close relationships to the well-known $k$-means algorithm and normalized cuts method.
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