SavedAdd your journal

Journals / Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics / 2021 / Cilt: 70 - Sayı: 2

A graph associated to a commutative semiring

Shahabaddin Ebrahimi ATANİ
Mehdi KHORAMDEL
Saboura DOLATI PISH HESARI
Mahsa NIKMARD ROSTAM ALIPOUR
Pages
984–996
Publication Date
2021-12-31
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Moreover, we characterize the planarity, chromatic number, clique number and independence number of GH(R)GH(R).

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Let RR be a commutative finite semiring with nonzero identity and HH be an arbitrary multiplicatively closed subset RR.
📊
Key Findings & Data: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+y∈Hx+y∈H.
💡
Conclusion & Impact:Moreover, we characterize the planarity, chromatic number, clique number and independence number of GH(R)GH(R).
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:I-semiring · planar graphs · clique number

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+y∈Hx+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).

🔖

Cite This Article

ATANİ, S. E., KHORAMDEL, M., HESARI, S. D. P., ALIPOUR, M. N. R. (2021). A graph associated to a commutative semiring. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Cilt: 70 - Sayı: 2, 984–996. https://ajindex.com/articles/746715

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (15)

  1. 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
  2. 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
  3. 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
  4. 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
  5. Beck, I., Coloring of commutative rings, J. Algebra, 116 (1988), 208-226. https://doi.org/10.1016/0021-8693(88)90202-5
  6. Bondy, J. A., Murty, U. S. R., Graph Theory, Graduate Texts in Mathematics, 244, Springer, New York, 2008.
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. Atani, S. E.