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A generalization of zero divisor graphs associated to commutative rings

  • M. Afkhami
  • A. Erfanian
  • K. Khashyarmanesh
  • N. Vaez Moosavi
Article
  • 42 Downloads

Abstract

Let R be a commutative ring with a nonzero identity element. For a natural number n, we associate a simple graph, denoted by \(\Gamma ^n_R\), with \(R^n\backslash \{0\}\) as the vertex set and two distinct vertices X and Y in \(R^n\) being adjacent if and only if there exists an \(n\times n\) lower triangular matrix A over R whose entries on the main diagonal are nonzero and one of the entries on the main diagonal is regular such that \(X^TAY=0\) or \(Y^TAX=0\), where, for a matrix \(B, B^T\) is the matrix transpose of B. If \(n=1\), then \(\Gamma ^n_R\) is isomorphic to the zero divisor graph \(\Gamma (R)\), and so \(\Gamma ^n_R\) is a generalization of \(\Gamma (R)\) which is called a generalized zero divisor graph of R. In this paper, we study some basic properties of \(\Gamma ^n_ R\). We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three.

Keywords

Zero divisor graph lower triangular matrix genus complete graph 

2010 Mathematics Subject Classification

15B33 05C10 05C25 05C45 

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Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  • M. Afkhami
    • 1
  • A. Erfanian
    • 2
  • K. Khashyarmanesh
    • 2
  • N. Vaez Moosavi
    • 2
  1. 1.Department of MathematicsUniversity of NeyshaburNeyshaburIran
  2. 2.Department of Pure MathematicsInternational Campus of Ferdowsi University of MashhadMashhadIran

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