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 $cd(G)={chi(1)‎ : ‎chiin Irr(G)}$‎. ‎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.