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

TL;DR

This paper identifies the non-cyclic nilpotent group of odd order with the maximum number of undirected edges in its directed power graph, extending previous results on cyclic groups.

Contribution

It determines the non-cyclic nilpotent group of a given odd order with the maximum undirected edges in its power graph and characterizes non-cyclic p-groups with maximum edges.

Findings

Identifies specific non-cyclic nilpotent groups with maximum undirected edges

Extends previous work from cyclic to non-cyclic groups

Provides characterization of maximum-edge non-cyclic p-groups

Abstract

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.