Lambda number of the enhanced power graph of a finite group

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∈⟨z⟩ for some z∈G. An L(2,1)-labeling of graph Γ is an integer labeling of V(Γ) 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 λ-number of Γ, denoted by λ(Γ), is the minimum range over all L(2,1)-labelings. In this article, we study the lambda number of the enhanced power graph PE(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 λ(PE(G))=n if and only if G is not a cyclic group of order n≥3. Finally, we compute the exact value of λ(PE(G)) if G is a finite nilpotent group.