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Some Properties of the Line Graphs Associated to the Total Graph of a Commutative Ring

Published: 2 April 2013
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Abstract

Let be a commutative ring with identity and its total graph. The subject of this article is the investigation of the properties of the corresponding line graph In particular, we determine the girth and clique number of In addition to that, we find the condition for to be Eulerian.

Published in Pure and Applied Mathematics Journal (Volume 2, Issue 2)
DOI 10.11648/j.pamj.20130202.11
Page(s) 51-55
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2013. Published by Science Publishing Group

Keywords

Total Graph; Line Graph; Commutative Ring

References
[1] I. Beck, "Coloring of commutative rings", J. Algebra 116, (1988) 208-226.
[2] D.F.Anderson, P.S. Livingston, "The zero-divisor graph of a commutative ring", J. Algebra 217, (1999), 434-447.
[3] D.F. Anderson, A. Badawi, "The total graph of a commutative ring", J. Algebra 2008, 320, 2706-2719.
[4] H.R. Maimani, C. Wickham, S. Yassemi, "Rings whose total graph have genus at most one", Rocky Mountain J. Math., to appear.
[5] H. Whitney, "Congruent graphs and connectivity of graphs", American Journal of Math. 54 (1932), 150-168.
[6] Hung-Jen Chiang-Hsieh, Pei-Feng Lee, Hsin-Ju Wang, "The embedding of line graphs associated to the zero-divisor graphs of commutative rings", Israel J. Math. 180, (2010) 193-222.
[7] Z. Petrović, Z. Pucanović, "The line graph associated to the total graph of a commutative ring", Ars Combinatoria, to appear.
[8] I. Kaplansky, " Commutative rings", Revised Edition, University of Chicago Press, Chicago, 1974.
[9] M.F. Atiyah, I.G. Macdonald, "Introduction to commutative algebra", Adison-Wesley Publishing Company, Reading, MA. 1969.
[10] D.B. West, "Introduction to graph theory", Prantice Hall, Inc., Upper Saddle River, NJ, 1996.
[11] R. Diestel, "Graph Theory", Third Edition, Graduate Texts in Mathematics 173, Springer-Verlag Berlin Heidelberg, 2005
Cite This Article
  • APA Style

    Aleksandra Lj. Erić, Zoran S. Pucanović. (2013). Some Properties of the Line Graphs Associated to the Total Graph of a Commutative Ring. Pure and Applied Mathematics Journal, 2(2), 51-55. https://doi.org/10.11648/j.pamj.20130202.11

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    ACS Style

    Aleksandra Lj. Erić; Zoran S. Pucanović. Some Properties of the Line Graphs Associated to the Total Graph of a Commutative Ring. Pure Appl. Math. J. 2013, 2(2), 51-55. doi: 10.11648/j.pamj.20130202.11

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    AMA Style

    Aleksandra Lj. Erić, Zoran S. Pucanović. Some Properties of the Line Graphs Associated to the Total Graph of a Commutative Ring. Pure Appl Math J. 2013;2(2):51-55. doi: 10.11648/j.pamj.20130202.11

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  • @article{10.11648/j.pamj.20130202.11,
      author = {Aleksandra Lj. Erić and Zoran S. Pucanović},
      title = {Some Properties of the Line Graphs Associated to the Total Graph of a Commutative Ring},
      journal = {Pure and Applied Mathematics Journal},
      volume = {2},
      number = {2},
      pages = {51-55},
      doi = {10.11648/j.pamj.20130202.11},
      url = {https://doi.org/10.11648/j.pamj.20130202.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20130202.11},
      abstract = {Let   be a commutative ring with identity and   its total graph. The subject of this article is the investigation of the properties of the corresponding line graph   In particular, we determine the girth and clique number of   In addition to that, we find the condition for   to be Eulerian.},
     year = {2013}
    }
    

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    T1  - Some Properties of the Line Graphs Associated to the Total Graph of a Commutative Ring
    AU  - Aleksandra Lj. Erić
    AU  - Zoran S. Pucanović
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    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
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    AB  - Let   be a commutative ring with identity and   its total graph. The subject of this article is the investigation of the properties of the corresponding line graph   In particular, we determine the girth and clique number of   In addition to that, we find the condition for   to be Eulerian.
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    IS  - 2
    ER  - 

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