Lambda Number of the enhanced power graph of a finite group

TL;DR

This paper investigates the lambda number of enhanced power graphs of finite groups, extending previous work on power graphs, and provides exact values for specific classes of groups such as simple and nilpotent groups.

Contribution

It extends the study of lambda numbers from power graphs to enhanced power graphs and characterizes these numbers for simple and nilpotent groups.

Findings

Lambda number equals group order for non-cyclic simple groups.

Lambda number is computed explicitly for finite nilpotent groups.

Extension of previous results from power graphs to enhanced power graphs.

Abstract

The enhanced power graph of a finite group $G$ is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. An $L( 2,1)$-labeling of graph $\Gamma$ is an integer labeling of $V(\Gamma)$ such that adjacent vertices have labels that differ by at least $2$ and vertices distance $2$ apart have labels that differ by at least $1$. The $\lambda$-number of $\Gamma$, denoted by $\lambda(\Gamma)$, is the minimum range over all $L( 2,1)$-labelings. In this article, we study the lambda number of the enhanced power graph $\mathcal{P}_E(G)$ of the group $G$. This paper extends the corresponding results, obtained in [22], of the lambda number of power graphs to enhanced power graphs. Moreover, for a non-trivial simple group $G$ of order $n$, we prove that $\lambda(\mathcal{P}_E(G)) = n$ if and only if $G$ is not a cyclic group of order $n\geq 3$. Finally, we compute the exact value of $\lambda(\mathcal{P}_E(G))$ if $G$ is a finite nilpotent group.