Bipartite divisor graph for the set of irreducible character degrees

‎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(۱)‎ : ‎\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\{۱\})‎, ‎such that an element p of \rho(G) is adjacent to an element m of cd(G)\setminus\{۱\} 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.

Keywords:

‎bipartite divisor graph‎ , ‎irreducible character degree‎ , ‎path‎ , ‎cycle