The simplicial rook graph ${\rm \mathcal{SR}}(m,n)$ is the graph whose vertices are vectors in $ \mathbb{N}^m$ such that for each vector the summation of its coordinates is $n$ and two vertices are adjacent if their corresponding vectors differ in exactly two coordinates. Martin and Wagner (Graphs Combin. (2015) 31:1589--1611) asked about the independence number of ${\rm \mathcal{SR}}(m,n)$ that is the maximum number of non attacking rooks which can be placed on a $(m-1)$-dimensional simplicial chessboard of side length $n+1$. In this work, we solve this problem and show that $\alpha({\rm \mathcal{SR}}(m,n))=\big(1-o(1)\big)\frac{\binom{n+m-1}{n}}{m}$. We also prove that for the domination number of rook graphs we have $\gamma({\rm \mathcal{SR}}(m, n))= \Theta (n^{m-2})$. Moreover we show that these graphs are Hamiltonian. The cyclic simplicial rook graph ${\rm \mathcal{CSR}}(m,n)$ is the graph whose vertices are vectors in $\mathbb{Z}^{m}_{n}$ such that for each vector the summation of its coordinates modulo $n$ is $0$ and two vertices are adjacent if their corresponding vectors differ in exactly two coordinates. In this work we determine several properties of these graphs such as independence number, chromatic number and automorphism group. Among other results, we also prove that computing the distance between two vertices of a given ${\rm \mathcal{CSR}}(m,n)$ is $ \mathbf{NP}$-hard in terms of $n$ and $m$.

翻译：简化的 rook 图形 $\ rm\ rm\ SR} (m,n) 是一个图表, 它的顶端是 $\ mathb{ n\ m 美元 的矢量, 所以对于每个矢量, 其坐标的加起来是 $n$, 而两个顶端是相邻的。 Martin and Wagner (Graphs Group. (2015) 31: 1589-1611) 询问 $(rm) 的独立的数 = rm = mathal{ {m, SR} (m,n) 美元, 它的顶端量是 不攻击的量 rook 量 $n 的矢量 。