Abstarct: Studying algebraic structures by assigning graphs to them and examining their properties has been an interesting research topic in recent decades that has raised interesting algebraic questions and results. In this thesis, we are interested to study and investigate the compressed zero-divisor graph of a ring R and some of its extensions. We first study the diameter of the compressed zero-divisor graph Γ_E (R), where R is a commutative ring. We give a complete characterization for the possible diameters of Γ_E (R) exclusively in terms of the ideals of R. Also, we extend the concept of compressed zero-divisor graph to non-commutatve case, and study the diameter of the zero-divisor and also the compressed zero-divisor graphs over skew Laurent polynomial rings Γ_E (R[x,x^(-1); α]) and over skew polynomial rings Γ_E (R[x,δ; α]), where R is reversible and (α,δ)-compatible. Moreover, we investigate the domination number of the zero-divisor graph Γ(R) the compressed zero-divisor graph Γ_E (R) and the unit graph G(R). Also, some relations between the domination number of Γ(R) and Γ(R[x,δ; α]), as well as the relations between the domination number of G(R) and G(R[[x; α]], are studied.