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Lambda Number of the enhanced power graph of a finite group | | Parveen
; Sandeep Dalal
; Jitender Kumar
; | | Date: |
1 Aug 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 $x, y
in langle z
angle$ 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 $ngeq 3$. Finally, we compute the exact value of
$lambda(mathcal{P}_E(G))$ if $G$ is a finite nilpotent group. | | Source: | arXiv, 2208.00611 | | Services: | Forum | Review | PDF | Favorites | |
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