Total graph of a commutative semiring with respect to singular ideal

Authors

DOI:

https://doi.org/10.22199/issn.0717-6279-2020-03-0032

Keywords:

Semiring, Total graph, Singular ideal, Induced subgraph

Abstract

Let S be a commutative semiring with unity. The singular ideal Z(S) of S is defined as Z(S) = {s ? S | sK = 0 for some essential ideal K of S}. In this paper, we introduce the notion of total graph of a commutative semiring with respect to the singular ideal. We define this graph as the undirected graph T(?(S)) with all elements of S as vertices and any two distinct vertices x and y are adjacent if and only if x + y ? Z(S). We discuss various characteristics of this total graph and also characterize some important properties of certain induced subgraphs of this total graph.

Author Biographies

Nabanita Goswami, Gauhati University.

Dept. of Mathematics.

Helen K. Saikia, Gauhati University.

Dept. of Mathematics.

References

A. Abbasi and S. Habibi, “The total graph of a commutative ring with respect to proper ideals”, Journal of the Korean Mathematical Society, vol. 49, no. 1, pp. 85–98, Jan. 2012, doi: 10.4134/JKMS.2012.49.1.085

S. Akbari and A. Mohammadian, “On the zero-divisor graph of a commutative ring”, Journal of algebra, vol. 274, no. 2, pp. 847–855, Apr. 2004, doi: 10.1016/S0021-8693(03)00435-6

D. F. Anderson and M. Naseer, “Beck?s coloring of a commutative ring”, Journal of algebra, vol. 159, no. 2, pp. 500–514, Aug. 1993, doi: 10.1006/jabr.1993.1171

D. F. Anderson, “On the diameter and girth of a zero-divisor graph, II”, Houston journal of mathematics, vol. 34, no. 2, pp. 361-371, 2008.

D. F. Anderson and A. Badawi, “The generalized total graph of a commutative ring”, Journal of algebra and its applications, vol. 12, no. 05, pp. 125–212, May 2013, doi: 10.1142/S021949881250212X

D. F. Anderson and A. Badawi, “The total graph of a commutative ring”, Journal of algebra, vol. 320, no. 7, pp. 2706–2719, Oct. 2008, doi: 10.1016/j.jalgebra.2008.06.028

D. F. Anderson and P. S. Livingston, “The zero-divisor graph of a commutative ring”, Journal of algebra, vol. 217, no. 2, pp. 434–447, Jul. 1999, doi: 10.1006/jabr.1998.7840

D. F. Anderson and S. Mulay, “On the diameter and girth of a zero-divisor graph”, Journal of pure and applied algebra, vol. 210, no. 2, pp. 543–550, Aug. 2007, doi: 10.1016/j.jpaa.2006.10.007

T. Asir and T. Tamizh Chelvam, “Genus of total graphs of commutative rings: a survey”, Electronic notes in discrete mathematics, vol. 63, pp. 59–68, Dec. 2017, doi: 10.1016/j.endm.2017.10.062

T. Asir and T. T. Chelvam, “On the total graph and its complement of a commutative ring”, Communications in algebra, vol. 41, no. 10, pp. 3820–3835, Oct. 2013, doi: 10.1080/00927872.2012.678956

S. E. Atani, “An ideal based zero-divisor graph of a commutative semiring”, Glasnik matematicki, vol. 44, no. 1, pp. 141–153, May 2009, doi: 10.3336/gm.44.1.07

S. E. Atani, “The zero-divisor graph with respect to ideals of a commutative semiring”, Glasnik matematicki, vol. 43, no. 2, pp. 309–320, Nov. 2008, doi: 10.3336/gm.43.2.06

S. E. Atani and S. Habibi, “The total torsion element graph of a module over a commutative ring”, Analele Universitatii "Ovidius" Constanta - Seria Matematica, vol. 19, no. 1, pp. 23-34, 2011. [On line]. Available: https://bit.ly/2z5V7kM

S. E. Atani, S. D. P. Hesari, and M. Khoramdel, “Total graph of a commutative semiring with respect to identity-summand elements”, Journal of the Korean Mathematical Society, vol. 51, no. 3, pp. 593–607, May 2014, doi: 10.4134/JKMS.2014.51.3.593

S. E. Atani, S. D. P. Hesari, and M. Khoramdel, “Total Identity-Summand Graph Of A Commutative Semiring With Respect To A Co-Ideal”, Journal of the Korean Mathematical Society, vol. 52, no. 1, pp. 159–176, Jan. 2015, doi: 10.4134/JKMS.2015.52.1.159

S. E. Atani and F. E. K. Saraei, “The total graph of a commutative semiring”, Analele Universitatii "Ovidius" Constanta - Seria Matematica, vol. 21, no. 2, pp. 21–33, Jun. 2013, doi: 10.2478/auom-2013-0021 [17] S. E. Atani and Z. E. Sarvandi, "The total graph of a commutative semiring with respect to proper ideals”, Journal of algebra and related topics, vol. 3, no. 2, pp. 27-41, 2015. [On line]. Available: https://bit.ly/3bM6X0E

I. Beck, “Coloring of commutative rings”, Journal of algebra, vol. 116, no. 1, pp. 208–226, Jul. 1988, doi: 10.1016/0021-8693(88)90202-5

B. Bolloba?s, Graph theory an introductory course. New York, NY: Springer, 1979, doi: 10.1007/978-1-4612-9967-7

J. S. Golan, Semirings and their applications. Dordrecht: Springer, 1999, doi: 10.1007/978-94-015-9333-5

J. Goswami, K. K. Rajkhowa, and H. K. Saikia, “Total graph of a module with respect to singular submodule”, Arab journal of mathematical sciences, vol. 22, pp. 242-249, 2016. [On line]. Available: https://bit.ly/3gbI5mu

M. H. Shekarriz, M. H. S. Haghighi, and H. Sharif, “On the total graph of a finite commutative ring”, Communications in algebra, vol. 40, no. 8, pp. 2798–2807, Aug. 2012, doi: 10.1080/00927872.2011.585680

Y. Talebi and A. Darzi, “The generalized total graph of a commutative semiring”, Ricerche di matematica, vol. 66, no. 2, pp. 579–589, Feb. 2017, doi: 10.1007/s11587-017-0321-4

H. S. Vandiver, “Note on a simple type of algebra in which the cancellation law of addition does not hold”, Bulletin of the American Mathematical Society, vol. 40, no. 12, pp. 914–921, Dec. 1934, doi: 10.1090/S0002-9904-1934-06003-8

Published

2020-06-03

How to Cite

[1]
N. Goswami and H. K. Saikia, “Total graph of a commutative semiring with respect to singular ideal”, Proyecciones (Antofagasta, On line), vol. 39, no. 3, pp. 517-527, Jun. 2020.

Issue

Section

Artículos