On the minimal (edge) connectivity of graphs and its applications to power graphs of finite groups

TL;DR

This paper provides a comprehensive characterization of finite groups based on the minimal edge connectivity of their power-related graphs, extending previous classifications and establishing new conditions for various group types.

Contribution

It introduces a necessary and sufficient condition for minimal edge connectivity in arbitrary graphs and applies this to classify finite groups via their enhanced and order superpower graphs.

Findings

Finite groups with minimally edge connected enhanced power graphs are characterized.

Finite p-groups are identified by minimally connected order superpower graphs.

All finite nilpotent groups with equal minimum degree and vertex connectivity in their order superpower graphs are characterized.

Abstract

In an earlier work, finite groups whose power graphs are minimally edge connected have been classified. In this article, first we obtain a necessary and sufficient condition for an arbitrary graph to be minimally edge connected. Consequently, we characterize finite groups whose enhanced power graphs and order superpower graphs, respectively, are minimally edge connected. Moreover, for a finite non-cyclic group $G$, we prove that $G$ is an elementary abelian $2$-group if and only if its enhanced power graph is minimally connected. Also, we show that $G$ is a finite $p$-group if and only if its order superpower graph is minimally connected. Finally, we characterize all the finite nilpotent groups such that the minimum degree and the vertex connectivity of their order superpower graphs are equal.