Enumerating conjugacy classes of graphical groups over finite fields

TL;DR

This paper introduces graph polynomials that count conjugacy classes of graphical groups over finite fields based on their sizes, linking graph theory and algebraic group properties.

Contribution

It presents a novel method for enumerating conjugacy classes of graphical groups over finite fields using specialized graph polynomials.

Findings

Defined new graph polynomials for enumeration

Established connections between graph structure and group conjugacy classes

Provided formulas for counting conjugacy classes over finite fields

Abstract

Each graph and choice of a commutative ring gives rise to an associated graphical group. In this article, we introduce and investigate graph polynomials that enumerate conjugacy classes of graphical groups over finite fields according to their sizes.