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A graph associated to centralizer of elements of a group

Department of Pure Mathematics, Faculty of Mathematical Sciences and Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran

E-mail Address: farangis.johari@mail.um.ac.ir

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and

Kazem Khashyarmanesh

https://orcid.org/0000-0003-3314-7298

Department of Pure Mathematics, Faculty of Mathematical Sciences and Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran

E-mail Address: khashyar@ipm.ir

Corresponding author.

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https://doi.org/10.1142/S1793557121501242Cited by:0 (Source: Crossref)

Abstract

For a given nonabelian finite group $G$ $G$ and $S \subseteq G ∖ Z (G),$ where $Z (G)$ denotes the center of $G,$ we introduce a new graph $Γ (G, S)$ associated to the group $G$ as follows: Take $G ∖ (S \cup Z (G))$ as its vertex set and two distinct vertices $x$ and $y$ being adjacent if and only if there exists an element $s \in S$ such that $[x, s] = 1 = [y, s] .$ This paper is devoted to investigate the properties of graphs $Γ (G, S)$ and establish some graph theoretical properties. Moreover, we describe the planarity of these graphs when $| S | = 1 .$ Also, we provide some examples of finite nonabelian groups $G$ with the property that if $Γ (G, S) ≅ Γ (H, S^{'})$ and $| S | = | S^{'} | = 1$ for some group $H$ and $S^{'} \subseteq H ∖ Z (H),$ then $| G | = | H | .$

Communicated by V. A. Artamonov

Keywords:

AMSC: 20D60, 05C25

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