Simple Groups Whose Prime Graph or Solvable Graph is Split

TL;DR

This paper proves that the compact form of the prime graph of any finite simple group is split, contributing to the understanding of the structural properties of graphs associated with finite groups.

Contribution

It establishes that the compact form of the prime graph for all finite simple groups is split, a new result in the study of group-related graphs.

Findings

Compact prime graphs of finite simple groups are split.

The splitness property holds for prime graphs and their compact forms.

Provides insights into the structure of graphs associated with finite simple groups.

Abstract

A graph is split if there is a partition of its vertex set into a clique and an independent set. The present paper is devoted to the splitness of some graphs related to finite simple groups, namely, prime graphs and solvable graphs, and their compact forms. It is proved that the compact form of the prime graph of any finite simple group is split.