Computing metric dimension of compressed zero divisor graphs associated to rings

For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph Γ_E(R) with vertex set Z(R_E) \ {[0]} = R_E \ {[0], [1]} defined by R_E = {[x] : x ∈ R}, where [x] = {y ∈ R : ann(x) = ann(y)} and the two distinct vertices [x] and [y] of Z(RE) are adjacent if and only if [x][y] = [xy] = [0], that is, if and only if xy = 0. In this paper, we study the metric dimension of the compressed zero divisor graph Γ_E(R), the relationship of metric dimension between Γ_E(R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of Γ_E(R). We provide a formula for the number of vertices of the family of graphs given by Γ_E(R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of Γ_E(R).