2024/01/09 by Mingzhu Chen, Ilya Gorshkov, Chen, Mingzhu +5
Engineering · Mathematics · #05C25 #20D60 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Group Theory (math.GR) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2401.04789
openalex publication_date 2024/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If G is a finite group, then the spectrum ω(G) is the set of all element orders of G. The prime spectrum π(G) is the set of all primes belonging to ω(G). A simple graph Γ(G) whose vertex set is π(G) and in which two distinct vertices r and s are adjacent if and only if rs ∈ ω(G) is called the Gruenberg-Kegel graph or the prime graph of G. In this paper, we prove that if G is a group of even order, then the set of vertices which are non-adjacent to 2 in Γ(G) form a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg-Kegel graph of a finite group. Besides this, we prove that a complete bipartite graph with each part of size at least 3 can not be isomorphic to the Gruenberg-Kegel graph of a finite group.