  Let $M$ be a module over a commutative ring $R$. The annihilating-submodule
graph of $M$, denoted by $AG(M)$, is a simple graph in which a non-zero
submodule $N$ of $M$ is a vertex if and only if there exists a non-zero proper
submodule $K$ of $M$ such that $NK=(0)$, where $NK$, the product of $N$ and
$K$, is denoted by $(N:M)(K:M)M$ and two distinct vertices $N$ and $K$ are
adjacent if and only if $NK=(0)$. This graph is a submodule version of the
annihilating-ideal graph and under some conditions, is isomorphic with an
induced subgraph of the Zariski topology-graph $G(\tau_T)$ which was introduced
in (The Zariski topology-graph of modules over commutative rings, Comm.
Algebra., 42 (2014), 3283--3296). In this paper, we study the domination number
of $AG(M)$ and some connections between the graph-theoretic properties of
$AG(M)$ and algebraic properties of module $M$.
