The intersection graph of a finite simple group has diameter at most 5

TL;DR

This paper proves that the intersection graph of any non-abelian finite simple group has a diameter at most 5, with only specific groups reaching this bound, thus answering a question from 2010.

Contribution

It establishes a universal upper bound of 5 for the diameter of intersection graphs of all non-abelian finite simple groups, resolving a previously open problem.

Findings

Diameter of intersection graphs is at most 5 for all such groups.

Only the baby monster and certain unitary groups attain diameter 5.

The result confirms the tightness of the bound for specific groups.

Abstract

Let $G$ be a non-abelian finite simple group. In addition, let $\Delta_G$ be the intersection graph of $G$, whose vertices are the proper nontrivial subgroups of $G$, with distinct subgroups joined by an edge if and only if they intersect nontrivially. We prove that the diameter of $\Delta_G$ has a tight upper bound of 5, thereby resolving a question posed by Shen (2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.