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 $B\left(G\right)$ for the set of irreducible complex character degrees $cd\left(G\right)$ is the undirected graph with vertex set consisting of the prime numbers dividing some element of $cd\left(G\right)$ and of the nonidentity character degrees in $cd\left(G\right)$, where a prime number $p$ is declared to be adjacent to a character degree $m$ if and only if $p$ divides $m$. The graph $B\left(G\right)$ 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 $B\left(G\right)$ 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.