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GENERALIZED CAYLEY GRAPH OF UPPER TRIANGULAR MATRIX RINGS

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

Abstract

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 ${\Gamma}^n_{T_n(R)}$.

Keywords

Cayley graph;matrix ring

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