• Afkhami, Mojgan ;
  • Hashemifar, Seyed Hosein ;
  • Khashyarmanesh, Kazem
  • Received : 2015.06.22
  • Published : 2016.07.31


Let R be a commutative ring with the non-zero identity and n be a natural number. ${\Gamma}^n_R$ is a simple graph with $R^n{\setminus}\{0\}$ as the vertex set and two distinct vertices X and Y in $R^n$ are adjacent if and only if there exists an $n{\times}n$ lower triangular matrix A over R whose entries on the main diagonal are non-zero such that $AX^t=Y^t$ or $AY^t=X^t$, where, for a matrix B, $B^t$ is the matrix transpose of B. ${\Gamma}^n_R$ is a generalization of Cayley graph. Let $T_n(R)$ denote the $n{\times}n$ upper triangular matrix ring over R. In this paper, for an arbitrary ring R, we investigate the properties of the graph ${\Gamma}^n_{T_n(R)}$.


Cayley graph;matrix ring


  1. M. Afkhami, S. Bahrami, K. Khashyarmanesh, and F. Shahsavar, The annihilating-ideal graph of a lattice, Georgian Math. J. 23 (2016), no. 1, 1-7.
  2. M. Afkhami, K. Khashyarmanesh, and Kh. Nafar, Generalized Cayley graphs associated to commutative rings, Linear Algebra Appl. 437 (2012), no. 3, 1040-1049.
  3. D. F. Anderson and A. Badawi, On the zero-divisor graph of a ring, Comm. Algebra 36 (2008), no. 8, 3073-3092.
  4. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier, New York, 1976.
  5. I. Chakrabarty, S. Ghosh, T. K. Mukherjee, and M. K. Sen, Intersection graphs of ideals of rings, Discrete Math. 309 (2009), no. 17, 5381-5392.
  6. A. V. Kelarev, Directed graphs and nilpotent rings, J. Austral. Math. Soc. Ser. A 65 (1998), no. 3, 326-332.
  7. A. V. Kelarev, On undirected Cayley graphs, Australas. J. Combin. 25 (2002), 73-78.
  8. A. V. Kelarev, Graph Algebras and Automata, Marcel Dekker, New York, 2003.
  9. A. V. Kelarev, Labelled Cayley graphs and minimal automata, Australas. J. Combin. 30 (2004), 95-101.
  10. A. V. Kelarev, On Cayley graphs of inverse semigroups, Semigroup Forum 72 (2006), no. 3, 411-418.
  11. A. V. Kelarev and C. E. Praeger, On transitive Cayley graphs of groups and semigroups, European J. Combin. 24 (2003), no. 1, 59-72.
  12. A. V. Kelarev and S. J. Quinn, A combinatorial property and power graphs of groups, Contributions to general algebra, 12 (Vienna, 1999), 229-235, Heyn, Klagenfurt, 2000.
  13. A. V. Kelarev and S. J. Quinn, Directed graphs and combinatorial properties of semigroups, J. Algebra 251 (2002), no. 1, 16-26.
  14. A. V. Kelarev, J. Ryan, and J. Yearwood, Cayley graphs as classifiers for data mining: the influence of asymmetries, Discrete Math. 309 (2009), no. 17, 5360-5369.
  15. T. Y. Lam, A First Course in Noncommutative Rings, 2nd Graduate tex in mathematics 131, Springer-Verlag, New York, 2001.
  16. H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947), 17-20.