On the Generation, Structure, and Symmetries of Minimal Prime Graphs

TL;DR

This paper advances the understanding of prime graphs of finite solvable groups by characterizing regular reseminant graphs, analyzing automorphisms, and introducing new construction methods for minimal prime graphs.

Contribution

It introduces new characterizations of regular reseminant graphs, studies automorphisms, and proposes novel methods for constructing minimal prime graphs beyond vertex duplication.

Findings

Characterization of regular reseminant graphs

Analysis of automorphisms of reseminant graphs

New construction methods for minimal prime graphs

Abstract

In this paper we continue the study of prime graphs of finite solvable groups. The prime graph, or Gruenberg-Kegel graph, of a finite group G has vertices consisting of the prime divisors of the order of G and an edge from primes p to q if and only if G contains an element of order pq. Since the discovery of a simple, purely graph theoretical characterization of the prime graphs of solvable groups in 2015 these graphs have been studied in more detail from a graph theoretic angle. In this paper we explore several new aspects of these graphs. We characterize regular reseminant graphs and study the automorphisms of reseminant graphs for arbitrary base graphs. We then study minimal prime graphs on larger vertex sets by a novel regular graph construction for base graphs and by proving results on prime graph properties under graph products. Lastly, we present the first new way, different from vertex duplication, to obtain a new minimal prime graph from a given minimal prime graph.