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A generalization of total graphs

  • M Afkhami
  • K Hamidizadeh
  • K Khashyarmanesh
Article
  • 44 Downloads

Abstract

Let R be a commutative ring with nonzero identity, \(L_{n}(R)\) be the set of all lower triangular \(n\times n\) matrices, and U be a triangular subset of \(R^{n}\), i.e., the product of any lower triangular matrix with the transpose of any element of U belongs to U. The graph \(GT^{n}_{U}(R^n)\) is a simple graph whose vertices consists of all elements of \(R^{n}\), and two distinct vertices \((x_{1},\dots ,x_{n})\) and \((y_{1},\dots ,y_{n})\) are adjacent if and only if \((x_{1}+y_{1}, \ldots ,x_{n}+y_{n})\in U\). The graph \(GT^{n}_{U}(R^n)\) is a generalization for total graphs. In this paper, we investigate the basic properties of \(GT^{n}_{U}(R^n)\). Moreover, we study the planarity of the graphs \(GT^{n}_{U}(U)\), \(GT^{n}_{U}(R^{n}{\setminus } U)\) and \(GT^{n}_{U}(R^n)\).

Keywords

Total graph triangular subset planarity girth diameter 

2000 Mathematics Subject Classification

05C10 05C25 13A15 

Notes

Acknowledgements

The authors are grateful to the referee for careful reading of the manuscript and helpful suggestions.

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Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of NeyshaburNeyshaburIran
  2. 2.Department of Pure MathematicsInternational Campus of Ferdowsi University of MashhadMashhadIran

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