Neighborhoods, connectivity, and diameter of the nilpotent graph of a finite group

TL;DR

This paper explores the properties of the nilpotent graph of finite groups, focusing on topological features, connectivity, and diameter bounds, especially for solvable groups and specific group classes.

Contribution

It characterizes finite solvable groups with nilpotent neighborhoods and provides bounds on the diameter of related graphs after removing universal vertices.

Findings

Characterization of finite solvable groups with nilpotent neighborhoods

Connectivity analysis of the reduced nilpotent graph

Upper bounds on the diameter of the graph for certain group classes

Abstract

The nilpotent graph of a group $G$ is the simple and undirected graph whose vertices are the elements of $G$ and two distinct vertices are adjacent if they generate a nilpotent subgroup of $G$. Here we discuss some topological properties of the nilpotent graph of a finite group $G$. Indeed, we characterize finite solvable groups whose closed neighborhoods are nilpotent subgroups. Moreover, we study the connectivity of the graph $\Gamma(G)$ obtained removing all universal vertices from the nilpotent graph of $G$. Some upper bounds to the diameter of $\Gamma(G)$ are provided when $G$ belongs to some classes of groups.