Bipartite divisor graph for the set of irreducible character degrees

TL;DR

This paper introduces a bipartite divisor graph based on the set of irreducible character degrees of a finite group, and uses its combinatorial properties to analyze the group's structure, especially when the graph forms a path or cycle.

Contribution

It defines a new bipartite divisor graph for character degrees and explores how its structure reveals properties of the finite group.

Findings

The bipartite divisor graph encodes prime divisibility relations among character degrees.

Structural properties of the graph, like being a path or cycle, relate to specific group characteristics.

Abstract

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 $cd(G)=\{\chi(1) : \chi\in Irr(G)\}$. 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 $\rho(G)\cup (cd(G)\setminus\{1\})$, such that an element $p$ of $\rho(G)$ is adjacent to an element $m$ of $cd(G)\setminus\{1\}$ 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.