  In this paper, we propose a new family of graphs, matrix graphs, whose vertex
set $\mathbb{F}^{N\times n}_q$ is the set of all $N\times n$ matrices over a
finite field $\mathbb{F}_q$ for any positive integers $N$ and $n$. And any two
matrices share an edge if the rank of their difference is $1$. Next, we give
some basic properties of such graphs and also consider two coloring problems on
them. Let $\chi'_d(N\times n, q)$ (resp. $\chi_d(N\times n, q)$) denote the
minimum number of colors necessary to color the above matrix graph so that no
two vertices that are at a distance at most $d$ (resp. exactly $d$) get the
same color. These two problems were proposed in the study of scalability of
optical networks. In this paper, we determine the exact value of
$\chi'_d(N\times n,q)$ and give some upper and lower bounds on $\chi_d(N\times
n,q)$.
