# A Graph Theoretic Analysis of Leverage Centrality

## TL;DR

This paper explores the mathematical properties of leverage centrality in graphs, analyzing its behavior across different graph families and revealing a surprising link to triangle numbers in Cartesian products of paths.

## Contribution

It provides a mathematical investigation of leverage centrality, including properties and computations for various graph families, highlighting a novel connection to triangle numbers.

## Key findings

- Leverage centrality properties are characterized mathematically.
- Explicit calculations for different graph families are provided.
- A surprising link between leverage centralities and triangle numbers is established.

## Abstract

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.