Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2016; 53(4): 1017-1031

Printed July 31, 2016

https://doi.org/10.4134/BKMS.b150487

Copyright © The Korean Mathematical Society.

Generalized Cayley graph of upper triangular matrix rings

Mojgan Afkhami, Seyed Hosein Hashemifar, and Kazem Khashyarmanesh

University of Neyshabur, International Campus of Ferdowsi University of Mashhad, International Campus of Ferdowsi University of Mashhad

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

MSC numbers: 05C69, 05C99, 13A99