On the Generation, Structure, and Symmetries of Minimal Prime Graphs
Ziyu Huang, Thomas Michael Keller, Shane Kissinger, Wen Plotnick, Maya, Roma

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…
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Taxonomy
TopicsFinite Group Theory Research · Synthesis and properties of polymers · Multicomponent Synthesis of Heterocycles
