Exact modularity of line graphs of complete graphs

Abstract

Modularity is designed to measure the strength of a division of a network into clusters. For $n\in \mathbb{N}$, consider a set $S$ of $n$ elements. Let $V$ be the set of all subsets of $S$ of size $r$. Consider the graph $G\left(n,r,s\right)$ with vertices $V$ and edges that connect two vertices if and only if their intersection has size $s$. In this article, we find the exact modularity of $G\left(n,2,1\right)$ for $n\ge 5$.