Semi Unit Graphs of Commutative Semi Rings

Yaqoub Ahmed, M. Aslam

Abstract


In this article, we introduce semi unit graph of semiring \(S\) denoted by \(\xi(S)\). The set of all elements of $S$ are vertices of this graph where distinct vertices \(x\) and \(y\) are adjacent if and only if \(x+y\) is a semiunit of \(S\). We investigate some of the properties and characterization results on connectedness, distance, diameter, girth, completeness and connectivity of \(\xi(S)\).

Keywords


Semirings; Semiunits; \(k\)-ideals; Graphs

Full Text:

PDF

References


P. J. Allen, A fundamental theorem of homomorphisms for semirings, Proc. Amer. Math. Soc. 21 (1969), 412 – 416, DOI: 10.1090/S0002-9939-1969-0237575-4.

D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative rings, J. Algebra 217 (1999), 434 – 447, DOI: 10.1006/jabr.1998.7840.

S. E. Atani, An ideal based zero divisor graph of commutative rings, Glasnik Math. 44 (2009), 141 – 153, DOI: 10.3336/gm.44.1.07.

S. E. Atani, The ideal theory in quotients of commutative semirings, Glasnik Math. Ser. III 42 (2007), 301 – 308, DOI: 10.3336/gm.42.2.05.

S. E. Atani, The zero-divisor graph with respect to ideals of a commutative semirings, Glasnik Math. 43 (2008), 309 – 320, DOI: 10.3336/gm.43.2.06.

S. E. Atani and R. Atani, Some remarks on partitioning semirings, An. St. Univ. Ovidius Constanta 18 (2010), 49 – 62, https://www.academia.edu/487767/.

R. E. Atani and S. E. Atani, Ideal theory in commutative semirings, Bul. Acad. Stiinte Repub. Mold. Mat. 2 (2008), 14 – 23, http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=basm&paperid=15&option_lang=eng.

I. Beck, Coloring of a commutative ring, J. Algebra 116 (1988), 208 – 226, DOI: 10.1016/0021-8693(88)90202-5.

J. S. Golan, Semi rings and Affine Equations over Them: Theory and Applications, Springer Science Å Business Media, Dordrecht (2003), DOI: 10.1007/978-94-017-0383-3.

J. S. Golan, Semirings and their Applications, Kluwer Academic Pub., Dordrecht (1999), DOI: 10.1007/978-94-015-9333-5.

J. S. Golan, The theory of semirings with applications in mathematics and theoretical computer Science, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific and Technical (1992), https://catalog.princeton.edu/catalog/724763.

V. Gupta and J. N. Chaudhari, Right pi-regular semirings, Sarajevo J. Math. 2 (2006), 3 – 9, http://www.anubih.ba/Journals/vol-2,no-1,y06/02vishnugupta1.pdf.

U. Hebisch and H. J. Weinert, Semirings: Algebraic Theory and Applications in Computer Science, Series in Algebra: Vol. 5, World Scientific Publishing Co. Ltd., Singapore (1993), DOI: 10.1142/3903.

O. Ore, Note on Hamilton circuits, American Mathematical Monthly 67(1) (1960), 55, OI: 10.2307/2308928.

M. K. Sen and M. R. Adhikari, On maximal k-ideals of simirings, Proc. Amer. Math. Soc. 118 (1993), 699 – 703, DOI: 10.1090/S0002-9939-1993-1132423-6.

W. D. Wallis, A Beginner’s Guide to Graph Theory, 2nd edition, Springer (2007), DOI: 10.1007/978-0-8176-4580-9.




DOI: http://dx.doi.org/10.26713%2Fcma.v10i3.1203

Refbacks

  • There are currently no refbacks.


eISSN 0975-8607; pISSN 0976-5905