AN OVERVIEW ON THE BIPARTITE DIVISOR GRAPH FOR THE SET OF IRREDUCIBLE CHARACTER DEGREES

Abstract

Let G be a finite group. The bipartite divisor graph B(G) for the set of irreducible complex character degrees cd(G) is the undirected graph with vertex set consisting of the prime numbers dividing some element of cd(G) and of the nonidentity character degrees in cd(G), where a prime number p is declared to be adjacent to a character degree m if and only if p divides m. The graph B(G) is bipartite and it encodes two of the most widely studied graphs associated to the character degrees of a finite group: the prime graph and the divisor graph on the set of irreducible character degrees. The scope of this paper is two-fold. We draw some attention to B(G) by outlining the main results that have been proved so far, see for instance [10; 11; 25; 26; 27]. In this process we improve some of these results.