The Lambda Number of the Power Graph of Finite Simple Groups

Abstract

For a finite group G, the power graph ΓG is a graph whose set of vertices is equal to G and two distinct elements of G are adjacent if and only if one of them is a positive integer power of the other. An L(2, 1)-labelling of ΓG is an integer-valued function defined on G so that the distance between the images of two adjacent vertices (resp. vertices with distance two) are at least two (resp. at least one). The lambda number λ(G) is defined to be the least difference between the largest and the smallest integer values assigned to the vertices of ΓG for all possible L(2, 1)-labellings of ΓG. It is known that λ(G) ≥ |G|. In this paper, we prove that if G is a finite simple group, then λ(G) = |G| except when G is cyclic of prime order. This settles a partial classification of finite groups G that achieve the lower bound on of lambda number.