Certain properties of the enhanced power graph associated with a finite group

Kumar, Jitender; Ma, Xuanlong; Parveen, None; Singh, Siddharth

doi:10.1007/s10474-023-01304-y

Cited by 5 publications

(3 citation statements)

References 20 publications

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“…Moreover, they are closely linked to automata theory [21]. There exist various types of graphs, including but not limited to commuting graphs of groups [4,11,17], power graphs of semigroups [12,23], groups [22], intersection power graphs of groups [6], enhanced power graphs of groups [1,7,24], and comaximal subgroup graphs [15]. These graphs have been established to explore the properties of algebraic structures using graph theory.…”

Section: Introductionmentioning

confidence: 99%

On the proper enhanced power graphs of finite nilpotent groups

Bera

Dey

2022

Journal of Group Theory

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For a group 𝐺, the enhanced power graph of 𝐺 is a graph with vertex set 𝐺 in which two distinct vertices x , y x,y are adjacent if and only if there exists an element 𝑤 in 𝐺 such that both 𝑥 and 𝑦 are powers of 𝑤. The proper enhanced power graph is the induced subgraph of the enhanced power graph on the set G ∖ S G\setminus S , where 𝑆 is the set of dominating vertices of the enhanced power graph. In this paper, we at first classify all nilpotent groups 𝐺 such that the proper enhanced power graphs are connected and calculate their diameter. We also explicitly calculate the domination number of the proper enhanced power graphs of finite nilpotent groups. Finally, we determine the multiplicity of the Laplacian spectral radius of the enhanced power graphs of nilpotent groups.

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Section: Introductionmentioning

confidence: 99%

On the proper enhanced power graphs of finite nilpotent groups

Bera

Dey

2022

Journal of Group Theory

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“…Furthermore, they are closely connected to automata theory [19]. Various types of graphs exist, including but not limited to commuting graphs of groups [3,8,16], power graphs of semigroups [9,21], groups [20], intersection power graphs of groups [4], enhanced power graphs of groups [1,5,22], and comaximal subgroup graphs [14]. These graphs have been established to investigate the properties of algebraic structures using graph theory.…”

Section: Introductionmentioning

confidence: 99%

“…[22, Theorem 5.6] Let G = P 1 × • • • × P r be a non-cyclic nilpotent group with r ≥ 2. Suppose that each Sylow subgroup of G is cyclic except P k for some k ∈ [r].…”

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confidence: 99%