  \begin{abstract} In this paper, we investigate a shift arising from graph
$G$. We prove that any $k$-dimensional shift of finite type can be generated
through a $k$-dimensional graph. We investigate the structure of the shift
space using the generating matrices for the shift space. We prove that a two
dimensional shift space has a horizontally (vertically) periodic point if and
only if it possesses a $(m,n)$-periodic point (for some $m,n\in
\mathbb{Z}\setminus \{0\}$). We prove that a shift space is finite if and only
if it can be generated by permutation matrices. We study the non-emptiness
problem and existence of periodic points in terms of the generating matrices.
