Lambda Number of the enhanced power graph of a finite group

arXiv2208.00611MaRDI QIDQ6406607

Jitender Kumar, Sandeep Dalal, Parveen

Publication date: 1 August 2022

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 $x,yinlanglezangle$ for some $zinG$. 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 $ngeq3$. Finally, we compute the exact value of $lambda(mathcal{P}_E(G))$ if $G$ is a finite nilpotent group.

Mathematics Subject Classification ID

Graphs and abstract algebra (groups, rings, fields, etc.) (05C25)

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