  Let $R$ be a commutative ring with identity. We define a graph
$\Gamma_{\aut}(R)$ on $ R$, with vertices elements of $R$, such that any two
distinct vertices $x, y$ are adjacent if and only if there exists $\sigma \in
\aut$ such that $\sigma(x)=y$. The idea is to apply graph theory to study orbit
spaces of rings under automorphisms. In this article, we define the notion of a
ring of type $n$ for $n\geq 0$ and characterize all rings of type zero. We also
characterize local rings $(R,M) $ in which either the subset of units ($\neq 1
$) is connected or the subset $M- \{0\}$ is connected in $\Gamma_{\aut}(R)$.
