# An overview on the bipartite divisor graph for the set of irreducible character degrees

## TL;DR

This paper reviews the bipartite divisor graph for irreducible character degrees of finite groups, highlighting key results, improvements, and open problems in understanding its structure and relation to other character degree graphs.

## Contribution

It provides an overview of existing results on the bipartite divisor graph, improves some findings, and discusses open problems in the study of this graph.

## Key findings

- Summarizes main results on the bipartite divisor graph.
- Improves certain known results about the graph.
- Identifies open problems for future research.

## Abstract

Let $G$ be a finite group. The bipartite divisor graph for the set of irreducible complex character degrees is the undirected graph with vertex set consisting of the prime numbers dividing some character degree and of the non-identity character degrees, where a prime number $p$ is declared to be adjacent to a character degree $m$ if and only if $p$ divides $m$. This graph 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 the bipartite divisor graph for the set of irreducible complex character degrees by outlining the main results that have been proved so far. In this process we improve some of these results and we leave some open problems.