Lambda number of the power graph of a finite group

TL;DR

This paper investigates the lambda number of power graphs of finite groups, establishing bounds, conditions for equality, and exact values for specific classes of groups such as dihedral, quaternion, and cyclic groups.

Contribution

It provides new bounds and characterizations for the lambda number of power graphs, including exact computations for several important classes of finite groups.

Findings

Established bounds for the lambda number of power graphs.

Derived necessary and sufficient conditions for bounds to be tight.

Computed exact lambda numbers for dihedral, quaternion, and certain cyclic groups.

Abstract

The power graph $\Gamma_G$ of a finite group $G$ is the graph with the vertex set $G$, where two distinct elements are adjacent if one is a power of the other. An $L(2, 1)$-labeling of a graph $\Gamma$ is an assignment of labels from nonnegative integers to all vertices of $\Gamma$ such that vertices at distance two get different labels and adjacent vertices get labels that are at least $2$ apart. The lambda number of $\Gamma$, denoted by $\lambda(\Gamma)$, is the minimum span over all $L(2, 1)$-labelings of $\Gamma$. In this paper, we obtain bounds for $\lambda(\Gamma_G)$, and give necessary and sufficient conditions when the bounds are attained. As applications, we compute the exact value of $\lambda(\Gamma_G)$ if $G$ is a dihedral group, a generalized quaternion group, a $\mathcal{P}$-group or a cyclic group of order $pq^n$, where $p$ and $q$ are distinct primes and $n$ is a positive integer.