Bipartite divisor graph for the set of irreducible character degrees

Abstract

Let $G$ be a finite group. We consider the set of the irreducible complex characters of $G$, namely $Irr(G)$, and the related degree set $cd(G)=\{\chi (1):\chi \in Irr(G)\}$. Let $\rho (G)$ be the set of all primes which divide some character degree of $G$. In this paper we introduce the bipartite divisor graph for $cd(G)$ as an undirected bipartite graph with vertex set $\rho (G)\cup (cd(G)\setminus \{1\})$, such that an element $p$ of $\rho (G)$ is adjacent to an element $m$ of $cd(G)\setminus \{1\}$ if and only if $p$ divides $m$. We denote this graph simply by $B(G)$. Then by means of combinatorial properties of this graph, we discuss the structure of the group $G$. In particular, we consider the cases where $B(G)$ is a path or a cycle.

References in corpus (1)

• Square character degree graphs yield direct products