On the enhanced power graph of a finite group

Panda, Ramesh Prasad; Dalal, S. S.; Kumar, Jitender

doi:10.1080/00927872.2020.1847289

Cited by 21 publications

(6 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This graph was introduced by Kelarev and Quinn [31] and investigated by several authors (see for example [8], [9], [10], [11], [13]). The edges of the enhanced power graph of G are the pairs {x, y} of distinct vertices such that x, y is cyclic, i.e., such that each of x and y is a power of some z ∈ G. This graph was introduced to interpolate between the power graph and the commuting graph, but has since been studied in its own right (see [12], [41], [49], [50]).…”

Section: Introductionmentioning

confidence: 99%

Finite groups satisfying the independence property

Freedman¹,

Lucchini²,

Nemmi³

et al. 2022

Preprint

View full text Add to dashboard Cite

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 is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups H contain an element s such that the maximal subgroups of H containing s, but not containing the socle of H, are pairwise non-conjugate.

show abstract

Section: Introductionmentioning

confidence: 99%

Finite groups satisfying the independence property

Freedman¹,

Lucchini²,

Nemmi³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Also, they supplied a characterization of finite nilpotent groups whose enhanced power graphs are perfect. Recently, Panda et al [21] studied the graph-theoretic properties viz. minimum degree, independence number, matching number, strong metric dimension and perfectness of enhanced power graph over finite abelian groups and some non abelian groups such as Dihedral groups, Dicyclic groups and the group U 6n .…”

Section: Introductionmentioning

confidence: 99%

On The Enhanced Power Graph of a Semigroup

Dalal¹,

Kumar²,

Singh³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The enhanced power graph Pe(S) of a semigroup S is a simple graph whose vertex set is S and two vertices x, y ∈ S are adjacent if and only if x, y ∈ z for some z ∈ S, where z is the subsemigroup generated by z. In this paper, first we described the structure of Pe(S) for an arbitrary semigroup S. Consequently, we discussed the connectedness of Pe(S). Further, we characterized the semigroup S such that Pe(S) is complete, bipartite, regular, tree and null graph, respectively. Also, we have investigated the planarity together with the minimum degree and independence number of Pe(S). The chromatic number of a spanning subgraph, viz. the cyclic graph, of Pe(S) is proved to be countable. At the final part of this paper, we construct an example of a semigroup S such that the chromatic number of Pe(S) need not be countable.2010 Mathematics Subject Classification. 20M10.

show abstract

“…Bera and Bhuniya [3] showed that there is a one-to-one correspondence between the maximal cliques in P e (G) and the maximal cyclic subgroups of G. In 2020, Zahirović, Bošnjak, and Madarász [32] showed that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups. Pandaab, Dalalc, and Kumarc [29] studied the minimum degree, independence number, and matching number of enhanced power graphs of finite groups. In [27] obtained an explicit formula for the metric dimension of an enhanced power graph.…”

Section: Introductionmentioning

confidence: 99%

Forbidden subgraphs in enhanced power graphs of finite groups

Lv²,

She

2021

Preprint

View full text Add to dashboard Cite

The enhanced power graph of a finite group is the graph whose vertex set is the set of the elements of this group, and two distinct vertices are adjacent if they generate a cyclic subgroup. In this paper, we classify all finite groups whose enhanced power graphs are split graphs and threshold graphs. We also classify all finite nilpotent groups whose enhanced power graphs are chordal graphs and Cographs. Finally, we give some families of non-nilpotent groups whose enhanced power graphs are chordal graphs and Cographs. The results partly answer a question was posed by Peter J. Cameron.

show abstract

On the enhanced power graph of a finite group

Cited by 21 publications

References 29 publications

Finite groups satisfying the independence property

Finite groups satisfying the independence property

On The Enhanced Power Graph of a Semigroup

Forbidden subgraphs in enhanced power graphs of finite groups

Contact Info

Product

Resources

About