Lambda number of the enhanced power graph of a finite group

Abstract

The enhanced power graph of a finite group [Formula: see text] is the simple undirected graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if [Formula: see text] for some [Formula: see text]. An [Formula: see text]-labeling of graph [Formula: see text] is an integer labeling of [Formula: see text] such that adjacent vertices have labels that differ by at least [Formula: see text] and vertices distance [Formula: see text] apart have labels that differ by at least [Formula: see text]. The [Formula: see text]-number of [Formula: see text], denoted by [Formula: see text], is the minimum range over all [Formula: see text]-labelings. In this paper, we study the lambda number of the enhanced power graph [Formula: see text] of the group [Formula: see text]. This paper extends the corresponding results, obtained in [X. Ma, M. Feng and K. Wang, Lambda number of the power graph of a finite group, J. Algebraic Combin. 53(3) (2021) 743–754], of the lambda number of power graphs to enhanced power graphs. Moreover, for a nontrivial simple group [Formula: see text] of order [Formula: see text], we prove that [Formula: see text] if and only if [Formula: see text] is not a cyclic group of order [Formula: see text]. Finally, we determine the lambda number of the enhanced power graphs of nilpotent groups.