Neighborhoods, connectivity, and diameter of the nilpotent graph of a finite group
Costantino Delizia, Michele Gaeta, Mark L. Lewis, Carmine Monetta
TL;DR
This paper investigates the nilpotent graph of a finite group G, focusing on the structure of Nil_G(x) neighborhoods and the induced graph Γ(G) after removing universal vertices. It characterizes solvable n-groups as Frobenius groups with nilpotent complements and connects Γ(G)'s connectivity to the quotient G/Z_infty(G), relating Γ(G) to the commuting graph Γ_comm(G). It then establishes general and case-specific diameter bounds for Γ(G), providing tighter results for special group classes and presenting an example with diameter 7, while highlighting open questions and potential refinements in the solvable setting.
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 $Γ(G)$ obtained removing all universal vertices from the nilpotent graph of $G$. Some upper bounds to the diameter of $Γ(G)$ are provided when $G$ belongs to some classes of groups.