Research ArticleNo Access

Domination number of graphs associated with rings

Ebrahim Hashemi

Faculty of Mathematical Sciences, Shahrood University of Technology, P. O. Box: 316-3619995161, Shahrood, Iran

E-mail Address: eb_hashemi@yahoo.com

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Mona Abdi

Faculty of Mathematical Sciences, Shahrood University of Technology, P. O. Box: 316-3619995161, Shahrood, Iran

E-mail Address: mona_abdi1368@yahoo.com

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Abdollah Alhevaz

Faculty of Mathematical Sciences, Shahrood University of Technology, P. O. Box: 316-3619995161, Shahrood, Iran

E-mail Address: a.alhevaz@gmail.com

Corresponding author.

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, and

Huadong Su

School of Mathematics and Statistics, Nanning Normal University, Nanning 530001, P. R. China

E-mail Address: huadongsu@sohu.com

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https://doi.org/10.1142/S0219498820500097Cited by:1 (Source: Crossref)

Abstract

The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. Let $R$ be an associative (not necessarily commutative) ring. We focus on the domination number of the zero-divisor graph $Γ (R)$ , the compressed zero-divisor graph $Γ_{E} (R)$ and the unit graph $G (R)$ . We find some relations between the domination number of the zero-divisor graph and that of the compressed zero-divisor graph. Moreover, some relations between the domination number of $Γ (R)$ and $Γ (R [x; α, δ])$ , as well as the relations between the domination number of $G (R)$ and $G (R [[x; α]])$ , are studied.

Communicated by T. H. Ha

Keywords:

AMSC: 16U99, 05C69, 16S36

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