  In 2010, Joyce et. al defined the leverage centrality of vertices in a graph
as a means to analyze functional connections within the human brain. In this
metric a degree of a vertex is compared to the degrees of all it neighbors. We
investigate this property from a mathematical perspective. We first outline
some of the basic properties and then compute leverage centralities of vertices
in different families of graphs. In particular, we show there is a surprising
connection between the number of distinct leverage centralities in the
Cartesian product of paths and the triangle numbers.

Combinatorics