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jzs-10786

Ideal Graphs Supported By Given Ideals of Commutative Rings


F. H. Abdulqadr1


1Mathematics Department, College of Education, Salahaddin University, Erbil-Iraq

Original: 9  November 2019       Revised: 12 January 2019      Accepted: 30 January 2020         Published online: 20 June 2020  


Doi Linkhttps://doi.org/10.17656/jzs.10786


Abstract

In this paper we introduce and study a new kind of graph that constructed by non-trivial ideals of a commutative ring with identity. Let R be a commutative ring with identity and P be a non-trivial ideal of R. The ideal graph supported by the ideal P, denoted by (P), is the undirected graph whose vertices are those non-trivial ideals I of R such that there exists a non-trivial ideal JI of R with IJP, and every two vertices I and J are adjacent if IJ and IJP. We investigate the connectivity, completeness and planarity of the graph (P). Also we explore the diameter, girth, domination, clique number and chromatic number of (P).


Key Words: Ideal graph supported by given ideals of commutative rings, connected graphs , Clique and chromatic number.

 
 References

[1] Akbari, S. and Mohamadian, A., "On the Zero-Divisor Graph of a Commutative Ring", Journal of Algebra. Vol. 274, No. 2, pp.847-855. (2004).

[2] Andersn, D. F. and  Livingston, P. S., "The zero divisor graph of a commutative ring", Journal of Algebra. Vol. 217, pp. 434-447. (1999).

[3] Beck, I., "Coloring of commutative rings", Journal of Algebra. 116, pp. 208–226. (1988).

[4] Behboodi, M. and Rakeei, Z., "The Annihilating-Ideal Graph of Commutative Rings I", Journal of Algebra and Appl., Vol.10, No.4, pp. 727-739. (2011).

[5] David, S. and Richard, M., "Abstract Algebra", Prentice-Hall Inc. U. S. A. (1991).

[6] Gary, C. and Linda, L., "Graphs and Digraphs", 2nd ed., Wadsworth and Brooks/Cole, California. (1986).

[7] Gupta, R, Sen, S. M. K. and Ghosh, S., " A variation of zero-divisor graphs", Discuss. Math. Gen. Algebra Appl. Vol. 35, No. 2, pp. 159–176. (2015).

[8] Mahadevi, P. and Babujee, J., " On Special Structures of Zero-Divisor Graphs", International Journal of Pure and Applied Mathematics. Vol. 119, No. 13, pp. 281-287. (2018)

[9] Patil, A., Waphare, B. N. and Joshi, V., "Perfect zero-divisor graphs", Discrete Math. Vol. 340, No. 4, pp. 740–745. (2017)

[10] Shuker, N. H.,  Mohammad, H. Q. and Luma, A. K., "Hosoya and Wiener Index of Zero-Divisor Graph of  ,"Zanco Journal of Pure and Applied Sciences. Vol. 31, No. 2, pp. 45-52. (2019).

 

  


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