Total graph of a commutative semiring with respect to singular ideal

• Goswami, Nabanita ^[1] ; Saikia, Helen K. ^[1]
  1. [1] Gauhati University

     Gauhati University

     India

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 39, Nº. 3, 2020, págs. 517-527
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-2020-03-0032
• Enlaces
  • Texto completo
• Resumen

  • 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.

• Referencias bibliográficas
  • A. Abbasi and S. Habibi, “The total graph of a commutative ring with respect to proper ideals”, Journal of the Korean Mathematical Society,...
  • 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,...
  • 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,...
  • 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:...
  • 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....
  • 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....
  • T. Asir and T. Tamizh Chelvam, “Genus of total graphs of commutative rings: a survey”, Electronic notes in discrete mathematics, vol. 63,...
  • 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....
  • 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:...
  • S. E. Atani, “The zero-divisor graph with respect to ideals of a commutative semiring”, Glasnik matematicki, vol. 43, no. 2, pp. 309–320,...
  • S. E. Atani and S. Habibi, “The total torsion element graph of a module over a commutative ring”, Analele Universitatii "Ovidius"...
  • S. E. Atani, S. D. P. Hesari, and M. Khoramdel, “Total graph of a commutative semiring with respect to identity-summand elements”, Journal...
  • 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...
  • S. E. Atani and F. E. K. Saraei, “The total graph of a commutative semiring”, Analele Universitatii "Ovidius" Constanta - Seria Matematica,...
  • 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,...
  • M. H. Shekarriz, M. H. S. Haghighi, and H. Sharif, “On the total graph of a finite commutative ring”, Communications in algebra, vol. 40,...
  • Y. Talebi and A. Darzi, “The generalized total graph of a commutative semiring”, Ricerche di matematica, vol. 66, no. 2, pp. 579–589, Feb....
  • 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...