Minimal Prime Graphs of Solvable Groups

TL;DR

This paper investigates the properties of minimal prime graphs of finite solvable groups, analyzing their diameters, Hamiltonian cycles, and self-complementarity using graph theory methods.

Contribution

It introduces new insights into minimal prime graphs, including their structural properties and a novel notion of minimality, expanding understanding in group and graph theory.

Findings

Minimal prime graphs have specific diameter and Hamiltonian cycle properties.

Certain minimal prime graphs are self-complementary.

A new concept of minimality for prime graphs is proposed and analyzed.

Abstract

We explore graph theoretical properties of minimal prime graphs of finite solvable groups. In finite group theory studying the prime graph of a group has been an important topic for the past almost half century. Recently prime graphs of solvable groups have been characterized in graph theoretical terms only. This now allows the study of these graphs with methods from graph theory only. Minimal prime graphs turn out to be of particular interest, and in this paper we pursue this further by exploring, among other things, diameters, Hamiltonian cycles and the property of being self-complementary for minimal prime graphs. We also study a new, but closely related notion of minimality for prime graphs and look into counting minimal prime graphs.