On the inclusion graphs of \(S\)acts
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 nontrivial 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.
Keywords
 \(S\)Act
 inclusion graph
 diameter
 girth
 domination number
MSC
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