图的特征向量、度离散度与主比的比较
Comparing Eigenvector and Degree Dispersion with the Principal Ratio of a Graph
摘要：
图的主比是图的主特征向量的最大项与最小项之比。由于主比比较的是主特征向量的极值，因此它对异常值很敏感。这对于从经验数据中得出的图形（网络）可能是有问题的。为了解释这一点，我们考虑主特征向量（和度向量）的色散。更准确地说，我们考虑上述向量的变异系数，即向量的标准差与均值的比值。我们展示了这两个统计量是如何被相同的本金比率函数所约束的。此外，这个界限是尖锐的正则图。本文的目的是证明主特征向量（和度向量）的变异系数在极限下可以收敛或发散到主比。在这样做的时候，我们找到一个图族（完全分裂图）的例子，其主比收敛到黄金比例。最后，我们与上述统计和有趣的性质的完全分裂图的极值图。
英文摘要：
The principal ratio of a graph is the ratio of the greatest and least entry of its principal eigenvector. Since the principal ratio compares the extreme values of the principal eigenvector it is sensitive to outliers. This can be problematic for graphs (networks) drawn from empirical data. To account for this we consider the dispersion of the principal eigenvector (and degree vector). More precisely, we consider the coefficient of variation of the aforementioned vectors, that is, the ratio of the vector's standard deviation and mean. We show how both of these statistics are bounded above by the same function of the principal ratio. Further this bound is sharp for regular graphs. The goal of this paper is to show that the coefficient of variation of the principal eigenvector (and degree vector) can converge or diverge to the principal ratio in the limit. In doing so we find an example of a graph family (the complete split graph) whose principal ratio converges to the golden ratio. We conclude with conjectures concerning extremal graphs of the aforementioned statistics and interesting properties of the complete split graph.
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