A Graphical Representation of Rings via Automorphism Groups

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 $ngeq 0$ and characterize all rings of type zero. We also characterize local rings $(R,M) $ in which either the subset of units ($ eq 1 $) is connected or the subset $M- {0}$ is connected in $Gamma_{aut}(R)$.