  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 $m$.
