On a Generalized Zero-divisor Graph of a Commutative Ring with Respect to an ideal

Authors

  • Priyanka Pratim Baruah Department of Mathematics, Girijananda Chowdhury Institute of Management and Technology, Guwahati 781017
  • Kuntala Patra Department of Mathematics, Gauhati University, Guwahati 781014

DOI:

https://doi.org/10.26713/cma.v8i3.355

Keywords:

Commutative ring, Ideal, Generalized zero-divisor graph, Diameter, Girth.

Abstract

In this paper, we generalize the notion of the ideal-based zero-divisor graph of a commutative ring. Let R be a commutative ring and let I be an ideal of R. Here, we define a generalized zero-divisor graph of R with respect to I and denote this graph by ΓIG(R). We show that ΓIG(R) is connected with diameter at most three. If ΓIG(R) has a cycle, we show that the girth of ΓIG(R) is at most four. Also, we investigate the existence of cut vertices of ΓIG(R). Moreover,we examine certain situations when ΓIG(R) is a complete bipartite graph.

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References

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Published

30-12-2017
CITATION

How to Cite

Baruah, P. P., & Patra, K. (2017). On a Generalized Zero-divisor Graph of a Commutative Ring with Respect to an ideal. Communications in Mathematics and Applications, 8(3), 333–343. https://doi.org/10.26713/cma.v8i3.355

Issue

Vol. 8 No. 3 (2017)

Section

Research Article

License

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