Minimal cut-sets in the power graphs of certain finite non-cyclic groups

TL;DR

This paper investigates the minimal cut-sets of power graphs of finite non-cyclic groups, especially nilpotent and abelian groups, revealing conditions for unique minimal cut-sets and calculating their vertex connectivity.

Contribution

It provides new results on the structure of minimal cut-sets in power graphs of certain finite non-cyclic groups, including conditions for uniqueness and connectivity.

Findings

Unique minimal cut-set under specified Sylow subgroup conditions

Determined vertex connectivity for certain finite abelian groups

Extended understanding of power graph connectivity in non-cyclic groups

Abstract

The power graph of a group is the simple graph with vertices as the group elements, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. We study (minimal) cut-sets of the power graph of a (finite) non-cyclic (nilpotent) group which are associated with its maximal cyclic subgroups. Let $G$ be a finite non-cyclic nilpotent group whose order is divisible by at least two distinct primes. If $G$ has a Sylow subgroup which is neither cyclic nor a generalized quaternion $2$-group and all other Sylow subgroups of $G$ are cyclic, then under some conditions we prove that there is only one minimum cut-set of the power graph of $G$. We apply this result to find the vertex connectivity of the power graphs of certain finite non-cyclic abelian groups whose order is divisible by at most three distinct primes.