K₄-free character graphs with diameter three

2020/06/27 by Mahdi Ebrahimi, Ebrahimi, Mahdi
Computer Science · Engineering · Mathematics · #05C12 #05C25 #20C15 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2006.15249

openalex publication_date 2020/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a finite group and let \rmIrr(G) be the set of all irreducible complex characters of G. Let \rmcd(G) be the set of all character degrees of G and denote by ρ(G) the set of primes which divide some character degrees in \rmcd(G). The character graph Δ(G) associated to G is a graph whose vertex set is ρ(G) and there is an edge between two distinct primes p and q if and only if the product pq divides some character degree of G. Suppose the character graph Δ(G) is K₄-free with diameter 3. In this paper, we show that |ρ(G)|≠ 5, if and only if G≅ J₁ × A, where J₁ is the first Janko's sporadic simple group and A is abelian.

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