Recent developments on the power graph of finite groups – a survey (original) (raw)

Some Results on the Power Graph of Groups

2013

The aim of this paper is to identify complete power graphs of groups and compute their clique number and show that power graphs are perfect. Moreover, automorphism of the power graph of cyclic groups is determined here.

On the Power Graph of a Finite Group

2012

Abstract. The power graph P (G) of a group G is the graph whose vertex set is the group elements and two elements are adjacent if one is a power of the other. In this paper, we consider some graph theoretical properties of a power graph P (G) that can be related to its group theoretical properties. As consequences of our results, simple proofs for some earlier results are presented.

On the Structure of the Power Graph and the Enhanced Power Graph of a Group

The Electronic Journal of Combinatorics

Let GGG be a group‎. ‎The power graph of GGG is a graph with the vertex‎ ‎set GGG‎, ‎having an edge between two elements whenever one is a power of the other‎. ‎We characterize nilpotent groups whose power graphs have finite independence number‎. ‎For a bounded exponent group‎, ‎we prove its power graph is a perfect graph and we determine‎ ‎its clique/chromatic number‎. ‎Furthermore‎, ‎it is proved that for every group GGG‎, ‎the clique number of the power graph of GGG is at most countably infinite‎. ‎We also measure how close the power graph is to the commuting graph by introducing a new graph which lies in between‎. ‎We call this new graph as the enhanced power graph‎. ‎For an arbitrary pair of these three graphs we characterize finite groups for which this pair of graphs are equal‎.

Some results on the power graphs of finite groups

ScienceAsia, 2015

We classify planar graphs and complete power graphs of groups and show that the only infinite group with a complete power graph is the Prüfer group p ∞. Clique and chromatic numbers and the automorphism group of power graphs are investigated. We also prove that the reduced power graph of a group G is regular if and only if G is a cyclic p-group or exp(G) = p for some prime number p.

Certain properties of the power graph associated with a finite group

Journal of Algebra and Its Applications, 2014

The power graph [Formula: see text] of a group G is a simple graph whose vertex-set is G and two vertices x and y in G are adjacent if and only if one of them is a power of the other. The subgraph [Formula: see text] of [Formula: see text] is obtained by deleting the vertex 1 (the identity element of G). In this paper, we first investigate some properties of the power graph [Formula: see text] and its subgraph [Formula: see text]. We next provide necessary and sufficient conditions for a power graph [Formula: see text] to be a strongly regular graph, a bipartite graph or a planar graph. Finally, we obtain some infinite families of finite groups G for which the power graph [Formula: see text] contains some cut-edges.

On the power graphs of certain finite groups

Linear and Multilinear Algebra, 2020

The power graph of a group is a graph, whose node set is and two distinct elements are adjacent if and only if one is an integral power of the other. A metric dimension of a graph , denoted by ψ() is the minimum cardinality of the resolving set of. In this context, we study distant properties and detour distant properties such as closure, interior, distance degree sequence and eccentric subgraphs of the power graphs of certain finite non-abelian groups. As a consequence, we figure out the metric dimension and resolving polynomial of power graphs for dihedral and generalized quaternion groups by using neighbourhood and twin sets.

On the Automorphisms Group of Finite Power Graphs

Facta Universitatis, Series: Mathematics and Informatics, 2021

The power graph of a group GGG is the graph with vertex set GGG,having an edge joining xxx and yyy whenever one is a power of theother. The purpose of this paper is to study the automorphismgroups of the power graphs of infinite groups.

On chordalness of power graphs of finite groups

2022

A graph is called chordal if it forbids induced cycles of length 4 or more. In this paper, we investigate chordalness of power graph of finite groups. In this direction we characterize direct product of finite groups having chordal power graph. We classify all simple groups of Lie type whose power graph is chordal. Further we prove that the power graph of a sporadic simple group is always non-chordal. We also show that almost all groups of order upto 47 has chordal power graph.

Generalized power graph of groups

2018

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.

On the Connectivity and Independence Number of Power Graphs of Groups

Graphs and Combinatorics, 2020

Let G be a group. The power graph of G is a graph with vertex set G in which two distinct elements x, y are adjacent if one of them is a power of the other. We characterize all groups whose power graphs have finite independence number, show that they have clique cover number equal to their independence number, and calculate this number. The proper power graph is the induced subgraph of the power graph on the set G-\{1\}$$G-{1}. A group whose proper power graph is connected must be either a torsion group or a torsion-free group; we give characterizations of some groups whose proper power graphs are connected.