Research Article
BibTex RIS Cite
Year 2021, Volume: 70 Issue: 2, 984 - 996, 31.12.2021
https://doi.org/10.31801/cfsuasmas.783398

Abstract

References

  • Afkhami, M., Barati, Z., Khashyarmanesh, K., A graph associated to a lattice, Ricerche Mat., 63 (2014), 67–78. https://doi.org/10.1007/s11587-013-0164-6
  • Anderson, D. F., Livingston, P. S., The zero-divisor graph of a commutative rings, J. Algebra, 217 (1999), 434-447. https://doi.org/10.1006/jabr.1998.7840
  • Anderson, D. F., Badawi, A., The total graph of a commutative ring, J. Algebra, 320(7) (2008), 2706–2719. https://doi.org/10.1016/j.jalgebra.2008.06.028
  • Barati, Z., Khashyarmanesh, K., Mohammadi, F., Nafar, Kh., On the associated graphs to a commutative ring, J. Algebra Appl., 11(2) (2012), 1250037 (17 pages). https://doi.org/10.1142/S0219498811005610
  • Beck, I., Coloring of commutative rings, J. Algebra, 116 (1988), 208-226. https://doi.org/10.1016/0021-8693(88)90202-5
  • Bondy, J. A., Murty, U. S. R., Graph Theory, Graduate Texts in Mathematics, 244, Springer, New York, 2008.
  • Atani, S. E., The ideal theory in quotients of commutative semirings, Glas. Math., 42 (2007), 301–308. https://doi.org/10.3336/gm.42.2.05
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., Strong co-ideal theory in quotients of semirings, J. of Advanced Research in Pure Math., 5(3) (2013), 19–32. https://doi.org/10.5373/jarpm.1482.061212
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., The identity-summand graph of commutative semirings, J. Korean Math. Soc., 51 (2014), 189–202. https://doi.org/10.4134/JKMS.2014.51.1.189
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., Total graph of a commutative semiring with respect to identity-summand elements, J. Korean Math. Soc., 51(3) (2014), 593– 607. https://doi.org/10.4134/JKMS.2014.51.3.593
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., Total identity-summand graph of a commutative semiring with respect to a co-ideal, J. Korean Math. Soc., 52(1) (2015), 159-176. https://doi.org/10.4134/JKMS.2015.52.1.159
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., A co-ideal based identity-summand graph of a commutative semiring, Comment. Math. Univ. Carolin., 56(3) (2015), 269–285. https://doi.org/10.14712/1213-7243.2015.124
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., A graph associated to proper nonsmall ideals of a commutative ring, Comment. Math. Univ. Carolin., 58(1) (2017), 1-12. https://doi.org/10.14712/1213-7243.2015.189
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., Sedghi Shanbeh Bazari, M., Total graph of a 0-distributive lattice, Categories and General Algebraic Structures with Applications, 9(1) (2018), 15-27. https://doi.org/10.29252/cgasa.9.1.15
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., Sedghi Shanbeh Bazari, M., A semi-prime filter based identity-summand graph of a lattice, LE Matematich, Vol. LXXIII (2018), 297–318. https://doi.org/10.4418/2018.73.2.5
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., Sarvandi, Z. E., Intersection graphs of co-ideals of semirings, Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, 68(1) (2019), 840–851. https://doi.org/10.31801/cfsuasmas.481603
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., On a graph of ideals of a commutative ring, Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, 68(2) (2019), 2283-2297. https://doi.org/10.31801/cfsuasmas.534944
  • Atani S. E., Esmaeili Khalil Saraei, F., The total graph of a commutative semiring, An. Stiint. Univ. Ovidius Constanta Ser. Mat., 21(2) (2013), 21-33. https://doi.org/10.2478/auom-2013-0021
  • Golan, J. S., Semirings and Their Applications, Kluwer Academic Publisher Dordrecht, 1999. https://doi.org/10.1007/978-94-015-9333-5

A graph associated to a commutative semiring

Year 2021, Volume: 70 Issue: 2, 984 - 996, 31.12.2021
https://doi.org/10.31801/cfsuasmas.783398

Abstract

Let RR be a commutative finite semiring with nonzero identity and HH be an arbitrary multiplicatively closed subset RR. The generalized identity-summand graph of RR is the (simple) graph GH(R)GH(R) with all elements of RR as the vertices, and two distinct vertices xx and yy are adjacent if and only if x+yHx+y∈H. In this paper, we study some basic properties of GH(R)GH(R). Moreover, we characterize the planarity, chromatic number, clique number and independence number of GH(R)GH(R).

References

  • Afkhami, M., Barati, Z., Khashyarmanesh, K., A graph associated to a lattice, Ricerche Mat., 63 (2014), 67–78. https://doi.org/10.1007/s11587-013-0164-6
  • Anderson, D. F., Livingston, P. S., The zero-divisor graph of a commutative rings, J. Algebra, 217 (1999), 434-447. https://doi.org/10.1006/jabr.1998.7840
  • Anderson, D. F., Badawi, A., The total graph of a commutative ring, J. Algebra, 320(7) (2008), 2706–2719. https://doi.org/10.1016/j.jalgebra.2008.06.028
  • Barati, Z., Khashyarmanesh, K., Mohammadi, F., Nafar, Kh., On the associated graphs to a commutative ring, J. Algebra Appl., 11(2) (2012), 1250037 (17 pages). https://doi.org/10.1142/S0219498811005610
  • Beck, I., Coloring of commutative rings, J. Algebra, 116 (1988), 208-226. https://doi.org/10.1016/0021-8693(88)90202-5
  • Bondy, J. A., Murty, U. S. R., Graph Theory, Graduate Texts in Mathematics, 244, Springer, New York, 2008.
  • Atani, S. E., The ideal theory in quotients of commutative semirings, Glas. Math., 42 (2007), 301–308. https://doi.org/10.3336/gm.42.2.05
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., Strong co-ideal theory in quotients of semirings, J. of Advanced Research in Pure Math., 5(3) (2013), 19–32. https://doi.org/10.5373/jarpm.1482.061212
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., The identity-summand graph of commutative semirings, J. Korean Math. Soc., 51 (2014), 189–202. https://doi.org/10.4134/JKMS.2014.51.1.189
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., Total graph of a commutative semiring with respect to identity-summand elements, J. Korean Math. Soc., 51(3) (2014), 593– 607. https://doi.org/10.4134/JKMS.2014.51.3.593
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., Total identity-summand graph of a commutative semiring with respect to a co-ideal, J. Korean Math. Soc., 52(1) (2015), 159-176. https://doi.org/10.4134/JKMS.2015.52.1.159
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., A co-ideal based identity-summand graph of a commutative semiring, Comment. Math. Univ. Carolin., 56(3) (2015), 269–285. https://doi.org/10.14712/1213-7243.2015.124
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., A graph associated to proper nonsmall ideals of a commutative ring, Comment. Math. Univ. Carolin., 58(1) (2017), 1-12. https://doi.org/10.14712/1213-7243.2015.189
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., Sedghi Shanbeh Bazari, M., Total graph of a 0-distributive lattice, Categories and General Algebraic Structures with Applications, 9(1) (2018), 15-27. https://doi.org/10.29252/cgasa.9.1.15
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., Sedghi Shanbeh Bazari, M., A semi-prime filter based identity-summand graph of a lattice, LE Matematich, Vol. LXXIII (2018), 297–318. https://doi.org/10.4418/2018.73.2.5
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., Sarvandi, Z. E., Intersection graphs of co-ideals of semirings, Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, 68(1) (2019), 840–851. https://doi.org/10.31801/cfsuasmas.481603
  • Atani, S. E., Hesari, S.D.P., Khoramdel, M., On a graph of ideals of a commutative ring, Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, 68(2) (2019), 2283-2297. https://doi.org/10.31801/cfsuasmas.534944
  • Atani S. E., Esmaeili Khalil Saraei, F., The total graph of a commutative semiring, An. Stiint. Univ. Ovidius Constanta Ser. Mat., 21(2) (2013), 21-33. https://doi.org/10.2478/auom-2013-0021
  • Golan, J. S., Semirings and Their Applications, Kluwer Academic Publisher Dordrecht, 1999. https://doi.org/10.1007/978-94-015-9333-5
There are 19 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Shahabaddin Ebrahimi Atani 0000-0003-0568-9452

Mehdi Khoramdel 0000-0003-0663-0356

Saboura Dolatı Pısh Hesarı This is me 0000-0001-8830-636X

Mahsa Nıkmard Rostam Alıpour This is me 0000-0003-3264-7936

Publication Date December 31, 2021
Submission Date August 21, 2020
Acceptance Date June 4, 2021
Published in Issue Year 2021 Volume: 70 Issue: 2

Cite

APA Atani, S. E., Khoramdel, M., Dolatı Pısh Hesarı, S., Nıkmard Rostam Alıpour, M. (2021). A graph associated to a commutative semiring. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70(2), 984-996. https://doi.org/10.31801/cfsuasmas.783398
AMA Atani SE, Khoramdel M, Dolatı Pısh Hesarı S, Nıkmard Rostam Alıpour M. A graph associated to a commutative semiring. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. December 2021;70(2):984-996. doi:10.31801/cfsuasmas.783398
Chicago Atani, Shahabaddin Ebrahimi, Mehdi Khoramdel, Saboura Dolatı Pısh Hesarı, and Mahsa Nıkmard Rostam Alıpour. “A Graph Associated to a Commutative Semiring”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70, no. 2 (December 2021): 984-96. https://doi.org/10.31801/cfsuasmas.783398.
EndNote Atani SE, Khoramdel M, Dolatı Pısh Hesarı S, Nıkmard Rostam Alıpour M (December 1, 2021) A graph associated to a commutative semiring. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70 2 984–996.
IEEE S. E. Atani, M. Khoramdel, S. Dolatı Pısh Hesarı, and M. Nıkmard Rostam Alıpour, “A graph associated to a commutative semiring”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 70, no. 2, pp. 984–996, 2021, doi: 10.31801/cfsuasmas.783398.
ISNAD Atani, Shahabaddin Ebrahimi et al. “A Graph Associated to a Commutative Semiring”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70/2 (December 2021), 984-996. https://doi.org/10.31801/cfsuasmas.783398.
JAMA Atani SE, Khoramdel M, Dolatı Pısh Hesarı S, Nıkmard Rostam Alıpour M. A graph associated to a commutative semiring. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70:984–996.
MLA Atani, Shahabaddin Ebrahimi et al. “A Graph Associated to a Commutative Semiring”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 70, no. 2, 2021, pp. 984-96, doi:10.31801/cfsuasmas.783398.
Vancouver Atani SE, Khoramdel M, Dolatı Pısh Hesarı S, Nıkmard Rostam Alıpour M. A graph associated to a commutative semiring. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70(2):984-96.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.