On a new extension of the zero-divisor graph (II)

We continue our study of the new extension of zero-divisor graph. We give a complete characterization for the possible diameters of $\widetilde{\Gamma}(R)$ and $\widetilde{\Gamma}(R[x_1,\dots,x_n])$, we investigate the relation between the zero-divisor graph, the subgraph of total graph on $Z(R)^{\star}$ and $\widetilde{\Gamma}(R)$ and we present some other properties of $\widetilde{\Gamma}(R)$.