Certain properties of the proper trivial intersection power graph associated with a finite group

Changliang Wang; Chunqiang Cui; Changliang Wang; Chunqiang Cui

doi:10.3934/era.2026230

Electronic Research Archive

2026, Volume 34, Issue 8: 5184-5199. doi: 10.3934/era.2026230

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Research article

Certain properties of the proper trivial intersection power graph associated with a finite group

Changliang Wang ,
Chunqiang Cui ^,

School of Computer Engineering, Zhanjiang University of Science and Technology, Zhanjiang 524094, China

Received: 15 April 2026 Revised: 05 June 2026 Accepted: 11 June 2026 Published: 17 June 2026

Let $ G $ be a group with identity $ e $. The proper trivial intersection (TI) power graph $ \mathcal{N}(G)^{\ast} $ defined on $ G $ is the undirected graph whose vertices are the elements of $ G\setminus\{e\} $, and two distinct vertices $ a, b $ are adjacent if $ \langle a\rangle\cap \langle b\rangle = \{e\} $. The purpose of this paper is to explore how the graph theoretical properties of $ \mathcal{N}(G)^{\ast} $ can affect the group theoretical properties of $ G $. In particular, this paper determines the domination number of a proper TI-power graph and classifies finite groups whose proper TI-power graph is planar.
- proper trivial intersection power graph,
- domination number,
- planar graph,
- finite group
Citation: Changliang Wang, Chunqiang Cui. Certain properties of the proper trivial intersection power graph associated with a finite group[J]. Electronic Research Archive, 2026, 34(8): 5184-5199. doi: 10.3934/era.2026230

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Abstract

Let $ G $ be a group with identity $ e $. The proper trivial intersection (TI) power graph $ \mathcal{N}(G)^{\ast} $ defined on $ G $ is the undirected graph whose vertices are the elements of $ G\setminus\{e\} $, and two distinct vertices $ a, b $ are adjacent if $ \langle a\rangle\cap \langle b\rangle = \{e\} $. The purpose of this paper is to explore how the graph theoretical properties of $ \mathcal{N}(G)^{\ast} $ can affect the group theoretical properties of $ G $. In particular, this paper determines the domination number of a proper TI-power graph and classifies finite groups whose proper TI-power graph is planar.

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