A survey on enhanced power graphs of finite groups

Ma, Xuanlong; Kelarev, AV; Yuan, Lin; Wang, Kaishun

doi:10.5614/ejgta.2022.10.1.6

Cited by 16 publications

(3 citation statements)

References 55 publications

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“…However, all the finite groups having genus one (or two) enhanced power graphs have not been classified so far. For a detailed list of results and open problems on enhanced power graphs of groups, we refer the reader to [27].…”

Section: Historical Background and Main Resultsmentioning

confidence: 99%

Nilpotent groups whose Difference graphs have positive genus

Parveen¹

2023

Preprint

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The power graph of a finite group G is a simple undirected graph with vertex set G and two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group G is a simple undirected graph whose vertex set is the group G and two vertices a and b are adjacent if there exists c ∈ G such that both a and b are powers of c. In this paper, we study the difference graph D(G) of a finite group G which is the difference of the enhanced power graph and the power graph of G with all isolated vertices removed. We characterize all the finite nilpotent groups G such that the genus (or cross-cap) of the difference graph D(G) is at most 2. 2020 Mathematics Subject Classification. 05C25. Key words and phrases. Enhanced power graph, power graph, nilpotent groups, genus and cross-cap of a graph.

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Section: Historical Background and Main Resultsmentioning

confidence: 99%

Nilpotent groups whose Difference graphs have positive genus

Parveen¹

2023

Preprint

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“…Literature is abound with numerous remarkable results concerning a number of constructions of graphs from rings, semigroups or groups and their applications, including automata theory, see for instance [1,11,19,28,29,30,31,36,43,45] and references therein. The intersection graph of a semigroup was introduced by Bosák [10] in 1964.…”

Section: Introductionmentioning

confidence: 99%

On the intersection ideal graph of semigroups

Baloda,

Kumar

2023

QRS

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The intersection ideal graph Γ(S) of a semigroup S is a simple undirected graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I, J are adjacent if and only if their intersection is nontrivial. In this paper, we investigate the connectedness of Γ(S). We show that if Γ(S) is connected, then the diameter of Γ(S) is at most two. Further, we classify the semigroups S in terms of their ideals such that the diameter of Γ(S) is two. We obtain the domination number, independence number, girth and the strong metric dimension of Γ(S). We have also investigated the completeness, planarity and perfectness of Γ(S). We show that if S is a completely simple semigroup, then Γ(S) is weakly perfect. More over, in this article, we give an upper bound of the chromatic number of Γ(S). Finally, if S is the union of n minimal left ideals, then we obtain the metric dimension and the automorphism group of Γ(S).

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“…Since that paper has been published, there has been an explosion in the number of papers published on the enhanced power graph, and we are not going to try to reference all of those papers. Also, there has been an expository paper that focuses just on enhanced power graphs of groups and it is worth consulting this paper [12].…”

Section: Introductionmentioning

confidence: 99%

The cyclic graph (deleted enhanced power graph) of a direct product

Costanzo

Lewis

Schmidt

et al. 2021

Involve

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Let G be a finite group and construct a graph ∆(G) by taking G \ {1} as the vertex set of ∆(G) and by drawing an edge between two vertices x and y if x, y is cyclic. Let K(G) be the set consisting of the universal vertices of ∆(G) along the identity element. For a solvable group G, we present a necessary and sufficient conditon for K(G) to be nontrivial. We also develop a connection between ∆(G) and K(G) when |G| is divisible by two distinct primes and the diameter of ∆(G) is 2.

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A survey on enhanced power graphs of finite groups

Abstract: We survey known results on enhanced power graphs of finite groups. Open problems, questions and suggestions for future work are also included.

Cited by 16 publications

References 55 publications

Nilpotent groups whose Difference graphs have positive genus

Nilpotent groups whose Difference graphs have positive genus

On the intersection ideal graph of semigroups

The cyclic graph (deleted enhanced power graph) of a direct product

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