# On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups

## TL;DR

This paper investigates the minimum degree, edge-connectivity, and connectivity of power graphs of various finite groups, establishing conditions under which these parameters are equal and determining minimum disconnecting sets.

## Contribution

It provides new results on the equality of connectivity and minimum degree for power graphs of finite groups, including necessary and sufficient conditions for cyclic groups.

## Key findings

- Minimum degree of power graphs for certain groups determined.
- Edge-connectivity equals minimum degree for these power graphs.
- Conditions for equality of connectivity and minimum degree established.

## Abstract

The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. In this paper, the minimum degree of power graphs of certain classes of cyclic groups, abelian $p$-groups, dihedral groups and dicyclic groups are obtained. It is ascertained that the edge-connectivity and minimum degree of power graphs are equal, and consequently the minimum disconnecting sets of power graphs of the aforementioned groups are determined. Then the equality of connectivity and minimum degree of power graphs of finite groups is investigated and in this connection, certain necessary conditions are produced. A necessary and sufficient condition for the equality of connectivity and minimum degree of power graphs of finite cyclic groups is obtained. Moreover, the equality is examined for the power graphs of abelian $p$-groups, dihedral groups and dicyclic groups.