  We show that the von Neumann entropy (from herein referred to as the von
Neumann index) of a graph's trace normalized combinatorial Laplacian provides
structural information about the level of centralization across a graph. This
is done by considering the Theil index, which is an established statistical
measure used to determine levels of inequality across a system of `agents',
e.g., income levels across a population. Here, we establish a Theil index for
graphs, which provides us with a macroscopic measure of graph centralization.
Concretely, we show that the von Neumann index can be used to bound the graph's
Theil index, and thus we provide a direct characterization of graph
centralization via the von Neumann index. Because of the algebraic similarities
between the bound and the Theil index, we call the bound the von Neumann Theil
index. %From an information theoretic perspective, the von Neumann Theil index
describes the mutual information between the quantum state corresponding to a
graph and the maximally mixed state (which corresponds to the complete graph).
We elucidate our ideas by providing examples and a discussion of different
$n=7$ vertex graphs. We also discuss how the von Neumann Theil index provides a
more comprehensive measure of centralization when compared to traditional
centralization measures, and when compared to the graph's classical Theil
index. This is because it more accurately accounts for macro-structural changes
that occur from micro-structural changes in the graph (e.g., the removal of a
vertex). Finally, we provide future direction, showing that the von Neumann
Theil index can be generalized by considering the R\'enyi entropy. We then show
that this generalization can be used to bound the negative logarithm of the
graph's Jain fairness index.
