Group With Maximum Undirected Edges in Directed Power Graph Among All Finite Non-Cyclic Nilpotent Groups

In [Curtin and Pourgholi, A group sum inequality and its application to power graphs, J. Algebraic Combinatorics, 2014], it is proved that among all directed power graphs of groups of a given order $n$, the directed power graph of cyclic group of order $n$ has the maximum number of undirected edges. In this paper, we continue their work and we determine a non-cyclic nilpotent group of an odd order $n$ whose directed power graph has the maximum number of undirected edges among all non-cyclic nilpotent groups of order $n$. We next determine non-cyclic $p$-groups whose undirected power graphs have the maximum number of edges among all groups of the same order.