Lambda number of the power graph of a finite group

Abstract

The power graph \(\Gamma _G\) of a finite group G is the graph with the vertex set G, where two distinct elements are adjacent if and only if one is a power of the other. An L(2, 1)-labeling of a graph \(\Gamma \) is an assignment of labels from nonnegative integers to all vertices of \(\Gamma \) such that vertices at distance two get different labels and adjacent vertices get labels that are at least 2 apart. The lambda number of \(\Gamma \), denoted by \(\lambda (\Gamma )\), is the minimum span or range over all L(2, 1)-labelings of \(\Gamma \). In this paper, we obtain bounds for \(\lambda (\Gamma _G)\) and give necessary and sufficient conditions when the bounds are attained. As applications, we compute the exact value of \(\lambda (\Gamma _G)\) if G is a dihedral group, a generalized quaternion group, a \({\mathcal {P}}\)-group or a cyclic group of order \(pq^n\), where p and q are distinct primes and n is a positive integer.