Domination number of total graph of module

Shariatnia, Abbas; Tehranian, Abolfazl

Domination number of total graph of module

Document Type : Research Paper

Authors

¹ Islamic Azad University, Tehran, Iran

² Islamic Azad University

Abstract

Let $R$ be a commutative ring and $M$ be an $R$-module with $T(M)$ as subset, the set of torsion elements. The total graph of the module denoted by $T(\Gamma(M))$, is the (undirected) graph with all elements of $M$ as vertices, and for distinct elements $n,m \in M$, the vertices $n$ and $m$ are adjacent if and only if $n+m \in T(M)$. In this paper we study the domination number of $T(\Gamma(M))$ and
investigate the necessary conditions for being $\mathbb{Z}_{n}$ as module over $\mathbb{Z}_{m}$ and we find the domination number of $T(\Gamma(\mathbb{Z}_{n}))$.

Keywords

References

[1] D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra 320 (2008), 2706{2719.
[2] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), 434{447.
[3] D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra 159 (1993), 500{514.
[4] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208{226.
[5] M. Axtell and J. Stickles, Zero-divisor graphs of idealizations, J. Pure Appl. Algebra, 204 (2006), 235{243.
[6] S. Ebrahimi Atani and S. Habibi The total torsion element graph of a module over a commutative ring, Analele Stiintice ale Universitatii Ovidius Constanta, 19( 1)(2011), 2334.
[7] M. R. Garey, D. S. Johnson, Computers and Intractability. A Guide to the Theory of NPCompleteness, A Series of Books in the Mathematical Sciences, W. H. Freeman and Co., San Francisco, Calif., 1979.
[8] T. W. Haynes, S. T. Hedetniemi, P. J. Slater, Fundamentals of Domination in Graphs, Monographs and Textbooks in Pure and Applied Mathematics, 208, Marcel Dekker, Inc., New York, 1998.
[9] T. W. Haynes, S. T. Hedetniemi, P. J. Slater (Editors), Domination in Graphs. Advanced Topics, Monographs and Textbooks in Pure and Applied Mathematics, 209, Marcel Dekker, Inc., New York, 1998.
[10] H. R. Maimani, C. Wickham, S. Yassemi, Rings whose total graphs have genus at most one, Rocky Mountain J. Math. 42 (2012), 1551{1560.
[11] M. H. Shekarriza, M. H. Shirdareh Haghighi and H. Sharif, On the Total Graph of a Finite Commutative Ring , Comm. Algebra 40(8) (2012), 2798{2807.