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

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Abstract

Let G be a group with identity e. The proper trivial intersection (TI) power graph $\mathcal{N}(G)^{*}$ defined on G is the undirected graph whose vertices are the elements of $G \backslash\{e\}$, 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 $\mathcal{N}(G)^{*}$ 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.

Keywords

proper trivial intersection power graph / domination number / planar graph / finite group

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