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Abstract 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 xy = 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).
Pirzada et al. (Sat,) studied this question.