A Study of the Total Graph

Let R be a commutative ring with $\Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $\T(\Gamma(R))$. It is the (undirected) graph with all elements of R as vertices, and for distinct $x, y\in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y\in\Z(R)$. We study the chromatic number and edge connectivity of this graph.