Comparing Eigenvector and Degree Dispersion with the Principal Ratio of a Graph

TL;DR

This paper investigates the relationship between the principal ratio of a graph and the dispersion of its principal eigenvector and degree vector, introducing bounds and exploring their behavior in specific graph families.

Contribution

It establishes bounds linking the principal ratio to the coefficient of variation of eigenvectors and degree vectors, and analyzes their convergence or divergence in graph limits, including the complete split graph.

Findings

Bounds on dispersion related to the principal ratio

Complete split graph's principal ratio converges to the golden ratio

Coefficient of variation can converge or diverge to the principal ratio

Abstract

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.