# On the inclusion graphs of $S$-acts

Volume 18, Issue 3, pp 357--363 Publication Date: July 22, 2018       Article History
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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.

### Keywords

• $S$-Act
• inclusion graph
• diameter
• girth
• domination number

•  20M30
•  16W22
•  05C12
•  05C69

### References

• [1] S. Akbari, M. Habibi, A. Majidinya, R. Manaviyat, The inclusion ideal graph of rings, Comm. Algebra, 43 (2015), 2457–2465.

• [2] D. F. Anderson, A. Badawi , The total graph of a commutative ring , J. Algebra, 320 (2008), 2706–2719.

• [3] J. Bosák, The graphs of semigroups, in: Theory of Graphs and its Application, 1964 (1964), 119–125.

• [4] B. Csákány, G. Pollák, The graph of subgroups of a finite group, Czechoslovak Math. J., 19 (1969), 241–247.

• [5] A. Das, Subspace inclusion graph of a vector space, Comm. Algebra, 44 (2016), 4724–4731.

• [6] A. Delfan, H. Rasouli, A. Tehranian, Intersection graphs associated with semigroup acts , , (submitted),

• [7] F. R. DeMeyer, T. McKenzie, K. Schneider, The zero-divisor graph of a commutative semigroup, Semigroup Forum, 65 (2002), 206–214.

• [8] P. Devi, R. Rajkumar, Inclusion graph of subgroups of a group, Cornell University Library, 2016 (2016), 22 pages.

• [9] A. A. Estaji, T. Haghdadi, A. A. Estaji , Zero divisor graphs for S-act, Lobachevskii J. Math., 36 (2015), 1–8.

• [10] M. Kilp, U. Knauer, A. V. Mikhalev, Monoids, Acts and Categories, Walter de Gruyter & Co., Berlin (2000)

• [11] H. R. Maimani, M. R. Pournaki, S. Yassemi, Weakly perfect graphs arising from rings , Glasg. Math. J., 52 (2010), 417–425.

• [12] R. Nikandish, M. J. Nikmehr, The intersection graph of ideals of $\mathbb{Z}_n$ is weakly perfect, Cornell University Library, 2013 (2013), 8 pages.

• [13] H. Rasouli, A. Tehranian, Intersection graphs of S-acts , Bull. Malays. Math. Sci. Soc., 38 (2015), 1575–1587.

• [14] M. R. Sorouhesh, H. Doostie, C. M. Campbell, A sufficient condition for coinciding the Green graphs of semigroups, J. Math. Computer Sci., 17 (2017), 216–219.

• [15] D. B. West, Introduction to Graph Theory, Prentice Hall , Upper Saddle River (2001)