The Minimal (Edge) Connectivity of Some Graphs of Finite Groups

TL;DR

This paper classifies finite groups based on the minimal edge connectivity and connectivity properties of various associated graphs, revealing structural insights into their algebraic and combinatorial characteristics.

Contribution

It provides a comprehensive classification of finite groups whose specific graphs are minimally edge connected or connected, including commuting, order-sum, non-inverse, and co-prime graphs.

Findings

Classified groups with minimally edge connected commuting, order-sum, and non-inverse graphs.

Identified groups where co-prime graphs are minimally connected.

Provided a complete classification for groups with these graph properties.

Abstract

In this paper, we classify all the finite groups $G$ such that the commuting graph $\Gamma_C(G)$, order-sum graph $\Gamma_{OS}(G)$ and non-inverse graph $\Gamma_{NI}(G)$ are minimally edge connected graphs. We also classify all the finite groups $G$ for that, these graphs are minimally connected. We also classify some groups for that the co-prime graph $\Gamma_{CP}(G)$ has minimal edge connectedness. In final part, we classify all the finite groups $G$ for that co-prime graph $\Gamma_{CP}(G)$ is minimally connected.