  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.
