论文标题
比较特征向量和程度分散与图的主比例
Comparing Eigenvector and Degree Dispersion with the Principal Ratio of a Graph
论文作者
论文摘要
图的主比例是其主要特征向量的最大和最少进入的比率。由于主比例比较主要特征向量的极端值，因此对异常值很敏感。这对于从经验数据中得出的图形（网络）可能是有问题的。为此，我们考虑主要特征向量（和学位向量）的分散。更确切地说，我们考虑上述向量变化的系数，即矢量标准偏差和平均值的比率。我们展示了这些统计数据中的两个统计数据如何通过主比率相同的功能来界定。此外，对于常规图，此界限是锋利的。本文的目的是表明，主要特征向量（和程度向量）的变化系数可以在极限内收敛或分歧为主比率。在此过程中，我们找到了一个图族的示例（完整的拆分图），其主比例会收敛到黄金比率。我们以有关上述统计数据的极端图和完整拆分图的有趣属性的猜想结束。
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.
