Characterizing finite nilpotent groups associated with a graph theoretic equality

TL;DR

This paper characterizes finite nilpotent groups based on the equality of vertex connectivity and minimum degree in their power graphs, linking group structure with graph-theoretic properties.

Contribution

It provides a complete characterization of finite nilpotent groups where the power graph's vertex connectivity equals its minimum degree, a novel connection between algebra and graph theory.

Findings

Identifies conditions under which power graphs of finite nilpotent groups have equal vertex connectivity and minimum degree.

Establishes a characterization that links group properties with specific graph invariants.

Enhances understanding of the interplay between group structure and graph connectivity measures.

Abstract

The power graph of a group is the simple graph whose vertices are the group elements and two vertices are adjacent whenever one of them is a positive power of the other. We characterize the finite nilpotent groups whose power graphs have equal vertex connectivity and minimum degree.