THE UNIT GRAPH OF A COMMUTATIVE SEMIRING

Document Type : Original Manuscript

Authors

1 Department of Mathematics, North-Eastern Hill University P.O. Box 793022, Shillong, India.

2 Department of Basic Sciences & Social Sciences, North-Eastern Hill University, P.O. Box 793022, Shillong, India.

3 Department of Mathematics, Gauhati University, P.O. Box 781014, Guwahati, India.

Abstract

Let S be a commutative semiring with unity and U(S) be the set of all units of S. The unit graph of S, denoted by G(S), is the undirected graph with vertex set S and two distinct vertices x and y are adjacent if and only if x + y ∈ U(S). In this article, we have investigated some properties of unit graph G(S) of S regarding completeness, bipartiteness, connectedness, diameter and girth. Finally, we find a necessary and sufficient condition for G(S) to be traversable.

Keywords


 1. Y. Ahmed and M. Aslam, Semi unit graphs of commutative semi rings, Comm. Math. and Appl., 10(3) (2019), 519–530.
2. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434–447.
3. A. Ashrafi, H. R. Maimani, M. R. Pournaki and S. Yassemi, Unit graphs associated with rings, Comm. Algebra, 38(8) (2010), 2851–2871.
4. R. E. Atani and S. E. Atani, Ideal theory in commutative semirings, Bul. Acad. Stiinte Repub. Mold. Mat., 2 (2008), 14–23.
5. S. E. Atani and F. E. Khalil Saraei, The total graph of commutative semiring, An. St. Univ. Ovidius Constanta, 21(2) (2013), 21–33.
6. S. E. Atani, M. Khoramdel and S. D. P. Hesari, A graph associated to a commutative semiring, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 70(2) (2021), 984–996.
7. I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208–226.
8. L. Boro, M. M. Singh and J. Goswami, Unit graph of the ring Zm × Zn, Lobachevskii J. Math., 43(2) (2022), 345–352.
9. R. Diestel, Graph Theory, Third Edition, Graduate Texts in Mathematics 173, Springer-Verlag Berlin Heidelberg, 2005.
10. D. Dol┼żan and P. Oblak, The total graphs of finite commutative semirings, Results Math., 72 (2017), 193–204.
11. J. S. Golan, Semirings and Their Applications, Kluwar Academic Publishers, Dordrecht, 1999.
12. J. Goswami and L. Boro, On the weakly nilpotent graph of a commutative semiring, Bol. Soc. Paran. Mat., 41 (2023), 1–10.
13. R. P. Grimaldi, Graphs from rings, Proceedings of the 20th Southeastern Conference on Combinatorics, Graph Theory and Computing, Congr. Numer., 71 (1990), 95–103.
14. F. Heydari and M. J. Nikmehr, Unit graph of a left Artinian ring, Acta Math. Hungarica, 139 (2013), 134–146.
15. H. R. Maimani, M. R. Pournaki and S. Yassemi, Necessary and sufficient conditions for unit graphs to be Hamiltonian, Pacific J. Math., 249(2) (2011), 419–429.
16. H. R. Maimani, M. R. Pournaki and S. Yassemi, Rings which are generated by their units: a graph theoretical approach, Elem. Math., 65 (2010), 17–25.
17. H. R. Maimani, M. R. Pournaki and S. Yassemi, Weakly perfect graphs arising from rings, Glasgow Math. Journal, 52 (2010), 417–425.
18. M. A. Shamla, On some types of ideals in semirings, Masters Thesis, The Islamic University of Gaza, 2008.
19. H. Su and Y. Wei, The diameter of unit graphs of rings, Taiwanese J. Math., 23(1) (2019), 1–10.
 20. H. Su and Y. Zhou, On the girth of the unit graph of a ring, J. Algebra Appl., 13 (2014), Article ID: 1350082.
21. Y. Talebi and A. Darzi, The generalized total graph of a commutative semiring, Ricerche Mat., 66 (2017), 579–589.