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  • 1 Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran
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Abstract

Let R be a commutative ring with non-zero identity and G be a multiplicative subgroup of U(R), where U(R) is the multiplicative group of unit elements of R. Also, suppose that S is a non-empty subset of G such that S−1={s−1sS}⫅S. Then we define Γ(R,G,S) to be the graph with vertex set R and two distinct elements x,yR are adjacent if and only if there exists sS such that x+syG. This graph provides a generalization of the unit and unitary Cayley graphs. In fact, Γ(R,U(R),S) is the unit graph or the unitary Cayley graph, whenever S={1} or S={−1}, respectively. In this paper, we study the properties of the graph Γ(R,G,S) and extend some results in the unit and unitary Cayley graphs.

  • [1] Akhtar, R. Boggess, M. Jackson-Henderson, T. Karpman, R. Boggess, M. Jimnez, I. Kinzel, A. Pritikin, D. 2009 On the unitary Cayley graph of a finite ring Electron. J. Combin. 16 #R117.

    • Search Google Scholar
    • Export Citation
  • [2] Anderson, D. F. Badawi, A. 2008 The total graph of a commutative ring J. Algebra 320 27062719 .

  • [3] Ashrafi, N. Maimani, H. R. Pournaki, M. R. Yassemi, S. 2010 Unit graphs associated with rings Comm. Algebra 38 28512871 .

  • [4] Atiyah, M. F. Macdonald, I. G. 1969 Introduction to Commutative Algebra Addison-Wesley Reading, MA.

  • [5] Cheyne, B., Gupta, V. and Wheeler, C., Hamilton cycles in addition graphs, Rose-Hulman Undergraduate Math. Journal, 1 (2003) (electronic).

    • Search Google Scholar
    • Export Citation
  • [6] Corbas, B. Williams, G. D. 2000 Rings of order p 5. Part I. Nonlocal rings J. Algebra 231 677690 .

  • [7] Diestel, R. 2005 Graph Theory Springer-Verlag New York.

  • [8] Godsil, C. Royle, G. 2001 Algebraic Graph Theory Springer-Verlag New York .

  • [9] Grynkiewicz, D. Lev, V. F. Serra, O. 2007 The connectivity of addition Cayley graphs Electron. Notes Discrete Math. 29 135139 .

  • [10] Lanski, C. Maróti, A. 2009 Rings elements as sums of units Cent. Eur. J. Math. 7 395399 .

  • [11] Raphael, R. 1974 Rings which are generated by their units J. Algebra 28 199205 .

  • [12] Sato, J. Baba, K. 2010 The chromatic number of the simple graph associated with a commutative ring Sci. Math. Jpn. 71 187194.

  • [13] Sharma, P. K. Bhatwadekar, S. M. 1995 A note on graphical representation of rings J. Algebra 176 124127 .

  • [14] West, D. B. 2001 Introduction to Graph Theory 2 Prentice-Hall.

  • Impact Factor (2019): 0.588
  • Scimago Journal Rank (2019): 0.489
  • SJR Hirsch-Index (2019): 38
  • SJR Quartile Score (2019): Q2 Mathematics (miscellaneous)
  • Impact Factor (2018): 0.538
  • Scimago Journal Rank (2018): 0.488
  • SJR Hirsch-Index (2018): 36
  • SJR Quartile Score (2018): Q2 Mathematics (miscellaneous)

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