Generalized power graph of groups

TL;DR

This paper introduces a generalized power graph for groups, focusing on vertices that generate proper subgroups and edges based on non-trivial intersections of cyclic subgroups, analyzing properties like completeness and planarity.

Contribution

It extends the classical power graph concept to a broader class of vertices and investigates its structural properties such as completeness and planarity.

Findings

The generalized power graph's completeness depends on specific group properties.

Planarity of the graph varies with the group's structure.

The paper characterizes conditions under which the graph exhibits these properties.

Abstract

The power graph of an arbitrary group $G$ is a simple graph with all elements of $G$ as its vertices and two vertices are adjacent if one is a positive power of another. In this paper, we generalize this concept to a graph whose vertices are all elements of $G$ that generate a proper subgroup of $G$ and two elements are adjacent if the cyclic subgroup generated by which have non-trivial intersections. We concentrate on completeness and planarity of this graph.