11 Citations
Notes on the diameter of the complement of the power graph of a finite group
- Mathematics
- 2021
We determine the diameter of every connected component of the complement of the power graph and the enhanced power graph of a finite group, which completely answers two questions by Peter J. Cameron.
A note on the complement of the power graph of a finite group
- Mathematics
- 2021
We determine the diameter of every connected component of the complement of the power graph of a finite group, which completely answers a question by Peter J. Cameron.
On groups with chordal power graph, including a classification in the case of finite simple groups
- Mathematics
- 2022
We prove various properties on the structure of groups whose power graph is chordal. Nilpotent groups with this property have been classified by Manna, Cameron and Mehatari [The Electronic Journal of…
Classification of non-solvable groups whose power graph is a cograph
- MathematicsJournal of Group Theory
- 2023
Abstract In recent work, Cameron, Manna and Mehatari have studied the finite groups whose power graph is a cograph, which we refer to as power-cograph groups. They classify the nilpotent groups with…
On chordalness of power graphs of finite groups
- Mathematics
- 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…
Forbidden Subgraphs of co-prime Graphs of finite Groups
- Mathematics
- 2023
For a finite group G the co-prime graph Γ( G ) is defined as a graph with vertex set G in which two distinct vertices x and y are adjacent if and only if gcd ( o ( x ) , o ( y )) = 1 where o ( x ) and…
Finite groups satisfying the independence property
- Mathematics
- 2022
We say that a finite group G satisfies the independence property if, for every pair of distinct elements x and y of G, either {x, y} is contained in a minimal generating set for G, or one of x and y…
On Difference of Enhanced Power Graph and Power Graph of a Finite Group
- Mathematics
- 2022
. The difference graph D ( G ) of a finite group G is the difference of enhanced power graph of G and power graph of G , with all isolated vertices are removed. In this paper we study the connectedness…
References
SHOWING 1-10 OF 28 REFERENCES
The power graph of a torsion-free group
- Mathematics
- 2017
The power graphP(G) of a group G is the graph whose vertex set is G, with x and y joined if one is a power of the other; the directed power graph$$\overrightarrow{P}(G)$$P→(G) has the same vertex…
Undirected power graphs of semigroups
- Mathematics
- 2009
The undirected power graph G(S) of a semigroup S is an undirected graph whose vertex set is S and two vertices a,b∈S are adjacent if and only if a≠b and am=b or bm=a for some positive integer m. In…
On the Connectivity and Independence Number of Power Graphs of Groups
- MathematicsGraphs Comb.
- 2020
All groups whose power graphs have finite independence number are characterized, and it is shown that they have clique cover number equal to their independence number, and this number is calculated.
On the Structure of the Power Graph and the Enhanced Power Graph of a Group
- MathematicsElectron. J. Comb.
- 2017
It is proved that for every group G, the clique number of the power graph of $G$ is at most countably infinite and the graph is called theenhanced power graph.
Graphs defined on groups
- Mathematics
- 2021
This paper concerns aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that, in particular, they are invariant under the action…
The power graph of a finite group, II
- Mathematics
- 2010
Abstract The directed power graph of a group G is the digraph with vertex set G, having an arc from y to x whenever x is a power of y; the undirected power graph has an edge joining x and y whenever…
On the punctured power graph of a finite group
- MathematicsAustralas. J Comb.
- 2015
The punctured power graph P (G) of a finite group G is the graph which has as vertex set the nonidentity elements of G, where two distinct elements are adjacent if one is a power of the other. We…
Forbidden Subgraphs of Power Graphs
- MathematicsElectron. J. Comb.
- 2021
The powergraph is always perfect; and the groups whose power graph is a threshold graph are determined completely.
Finite groups with nilpotent centralizers
- Mathematics
- 1961
Introduction. The purpose of this paper is to clarify the structure of finite groups satisfying the following condition: (CN): the centralizer of any nonidentity element is nilpotent. Throughout this…
On a class of doubly transitive groups
- Mathematics
- 1972
THE class u(u> 3) of a doubly transitive group of degree n is, according to Bochert,f greater than \n — § Vw. If we confine our attention however to those doubly transitive groups in which one of the…