Non-torsion element graph of a module over a commutative ring
Résumé
Let R be a commutative ring with unity and M be a unitary R-module. Let T(M) be the set of all torsion elements of M and NT(M) = M − T(M) be the set of all non-torsion elements of M. The non-torsion element graph of M over R is an undirected simple graph GNT (M) with NT(M) as vertex set and any two distinct vertices x and y are adjacent if and only if x + y ∈ T(M). In this paper, we study the basic properties of the graph GNT (M). We also study the diameter and girth of GNT (M). Further, we determine the domination number and the bondage number of GNT (M). We establish a relation between diameter and domination number of GNT (M). We also establish a relation between girth and bondage number of GNT (M). We also establish a relation between girth and bondage number of GNT (M).
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Références
Akbari S., Kiani D., Mohammadi F. and Moradi S., The total graph and regular graph of a commutative ring , Journal of Pure and Applied Algebra , 213, 2224-2228, 2009.
Anderson D.D. and Naseer M., Beck’s coloring of a commutative ring , Journal of Algebra , 159, 500-514, 1993.
Anderson D.F. and Badawi A., The total graph of a commutative ring , Journal of Algebra , 320, 2706-2719, 2008.
Anderson D.F. and Livingston P.S., The zero-divisor graph of a commutative ring , Journal of Algebra , 217, 434-447, 1999.
Anderson F. W. and Fuller K. R., Rings and Categories of Modules , 2nd Edition, Graduate Texts in Mathematics 13, Springer-Verlag New York, Inc. 1992.
Atani S.E. and Habibi S., The total torsion element graph of a module over a commutative ring , An. St. Univ. Ovidius Constanta , Vol. 19(1), 23-34, 2011.
Beck I., Coloring of commutative rings , Journal of Algebra , 116, 208-226, 1988.
Chartrand G. and Zhang P., Introduction to graph theory , Tata McGraw-Hill, 2006.
Goswami J., Some domination properties of the total graph of a module with respect to singular submodule , Online Journal of Analytic Combinatorics , Issue 15 , 2020.
Goswami J., Rajkhowa K.K. and Saikia H.K., Total graph of a module with respect to singular submodule , Arab Journal of Mathematical Sciences , 22, 242-249, 2016.
Goswami J. and Saikia H.K., On the Line graph associated to the Total graph of a module, MATEMATIKA, Vol. 31, No. 1, 7-13, 2015.
Goswami J. and Shabani M., Domination in the entire nilpotent element graph of a module over a commutative ring , Proyecciones Journal of Mathematics, Vol. 40, No 6,1411-1430, December 2021.
Haynes T.W., Hedetniemi S.T. and Slatar P.J., Fundamental of domination in graphs, Marcel Dekker. Inc., 1998.
Haynes T.W., Hedetniemi S.T. and Slatar P.J., Domination in graphs-Advanced topics, Marcel Dekker. Inc., 2000.
Lam T. Y., Lectures on modules and rings , Graduate Texts in Mathematics 189, Springer-Verlag New York, Inc. 1999.
Lambeck J., Lectures on rings and modules , Blaisdell Publishing Company, Waltham, Toronto, London, 1966.
Mojdeh D. A. and Rahimi A. M., Dominating sets of some graphs associated to commutative rings , Communications in Algebra , 40, 3389-3396, 2012.
Patwari D., Saikia H.K. and Goswami J., Some results on domination in the generalized total graph of a commutative ring , Journal of Algebra and Related Topics, Vol. 10, No. 1, 119-128, 2022.
Shariatinia A. and Tehranian A., Domination number of total graph of module , Journal of Algebraic Structures and Their Applications , Vol. 2, No. 1, 1-8, 2015.
Shekarriz M. H., Haghighi M. H. S. and Sharif H., On the total graph of a finite commutative ring , Communications in Algebra , 40, 2798-2807, 2012.
Tamizh Chelvam T. and Asir T., Domination in the total graph of a commutative ring , Journal of Combinatorial Mathematics and Combinatorial Computing , 87, 147-158, 2013.
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