We shall investigate $\gamma(R)$, the zero-divisor graph of a commutative ring $R$ (where $0 \ne 1$). Specifically, if $R$ is the given ring, let $\gamma(R) = (V,E)$ with $V = Z(R)$ and $E=\{ \{x,y\} | xy=0\}$. We will show elementary properties of $\gamma(R)$ and restrict our attention to graphs with finitely many vertices.