Centralizers in finite groups and Domination number of their commuting graphs

TL;DR

This paper investigates the domination and total domination numbers of proper commuting graphs of finite groups, providing bounds, exact formulas for nilpotent groups, and specific results for well-known group families.

Contribution

It offers new bounds and exact formulas for domination numbers in finite groups' commuting graphs, especially for nilpotent groups and specific group families.

Findings

Established general bounds for domination number.

Derived exact formulas for nilpotent groups' commuting graphs.

Determined domination numbers for several well-known finite groups.

Abstract

The proper commuting graph $\mathcal{C}^{**}(G)$ of a finite group $G$ is the simple graph whose vertices are the noncentral elements of $G$ and two distinct vertices are adjacent if they commute. In this paper, we study the domination number and total domination number of proper commuting graphs of finite groups. We first obtain general bounds for the domination number of proper commuting graphs. For finite nilpotent groups, we exploit a strong product decomposition of commuting graphs to derive exact formulas for the domination number. We further determine the exact domination number and total domination number for proper commuting graphs of several well-known families of finite groups, connecting with the centralizers of those groups.