Generalized Cayley graph of upper triangular matrix rings
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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 $\Gamma ^n_{T_n(R)}$.
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 $\Gamma ^n_{T_n(R)}$.
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