直径为3的无$K_4$的character图
$K_4$-free character graphs with diameter three
摘要：
设$G$是一个有限群，$\rm{Irr}(G)$是$G$的所有不可约复特征标的集合.设$\rm{cd}(G)$是$G$的所有特征标度的集合，用$ρ(G)$表示除$\rm{cd}(G)$中某些特征标度的素数的集合.与$G$相关联的特征图$Δ(G)$是顶点集为$ρ(G)$且两个不同素数$p$和$q$之间存在边的图当且仅当乘积$pq$整除$G$的某个特征度数。假设特征图$Δ(G)$是直径为$3$的无$K_4$的.本文证明了$|ρ(G)|\neq 5$当且仅当$G\cong J_1 \times A$，其中$J_1$是第一个Janko散在单群，$A$是交换群.
英文摘要：
Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible complex characters of $G$. Let $\rm{cd}(G)$ be the set of all character degrees of $G$ and denote by $ρ(G)$ the set of primes which divide some character degrees in $\rm{cd}(G)$. 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 $K_4$-free with diameter $3$. In this paper, we show that $|ρ(G)|\neq 5$, if and only if $G\cong J_1 \times A$, where $J_1$ is the first Janko's sporadic simple group and $A$ is abelian.
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