On the inclusion graphs of \(S\)-acts

Volume 18, Issue 3, pp 357--363 http://dx.doi.org/10.22436/jmcs.018.03.10

Publication Date: July 22, 2018 Submission Date: November 23, 2017 Revision Date: January 10, 2018 Accteptance Date: January 30, 2018

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Authors

Abdolhossein Delfan - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran Hamid Rasouli - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran Abolfazl Tehranian - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

Abstract

In this paper, we define the inclusion graph \({\Bbb{Inc}}(A)\) of an \(S\)-act \(A\) which is a graph whose vertices are non-trivial subacts of \(A\) and two distinct vertices \(B_1,B_2\) are adjacent if \(B_1 \subset B_2\) or \(B_2 \subset B_1\). We investigate the relationship between the algebraic properties of an \(S\)-act \(A\) and the properties of the graph \(\Bbb{Inc}(A)\). Some properties of \(\Bbb{Inc}(A)\) including girth, diameter and connectivity are studied. We characterize some classes of graphs which are the inclusion graphs of \(S\)-acts. Finally, some results concerning the domination number of such graphs are given.

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Keywords

• \(S\)-Act
• inclusion graph
• diameter
• girth
• domination number

MSC

•  20M30
•  16W22
•  05C12
•  05C69

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