On bipartite divisor graph for character degrees

Abstract

‎‎The concept of the bipartite divisor graph for integer subsets has been considered in [M‎. ‎A‎. ‎Iranmanesh and C‎. ‎E‎. ‎Praeger‎, ‎Bipartite divisor graphs for integer subsets‎, Graphs Combin.‎,  26 (2010) 95--105.]‎. ‎In this paper‎, ‎we will consider this graph for the set of character degrees of a finite group $G$ and obtain some properties of this graph‎. ‎We show that if $G$ is a solvable group‎, ‎then the number of connected components of this graph is at most $2$ and if $G$ is a non-solvable group‎, ‎then it has at most $3$ connected components‎. ‎We also show that‎ ‎the diameter of a connected bipartite divisor graph is bounded by $7$ and obtain some properties of groups whose graphs are complete bipartite graphs‎.