  For a simple graph $G$ with vertex set $V(G)=\{v_1,...,v_n\}$, we define the
closed neighborhood set of a vertex $u$ as $N[u]= \{v \in V(G) \; | \; v \;
\text{is adjacent to} \; u \; \text{or} \; v=u \}$ and the closed neighborhood
matrix $N(G)$ as the matrix whose $i$th column is the characteristic vector of
$N[v_i]$. We say a set $S$ is odd dominating if $N[u]\cap S$ is odd for all
$u\in V(G)$. We prove that the parity of the cardinality of an odd dominating
set of $G$ is equal to the parity of the rank of $G$, where rank of $G$ is
defined as the dimension of the column space of $N(G)$. Using this result we
prove several corollaries in one of which we obtain a general formula for the
nullity of the join of graphs.
