<abstract xmlns="http://www.w3.org/1999/xhtml">For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph ΓE(R) with vertex set Z(RE) \ {[0]} = RE \ {[0], [1]} defined by RE = {[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 ΓE(R), the relationship of metric dimension between ΓE(R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of ΓE(R). We provide a formula for the number of vertices of the family of graphs given by ΓE(R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of ΓE(R).</abstract>

For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph ΓE(R) with vertex set Z(RE) \ {[0]} = RE \ {[0], [1]} defined by RE = {[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 ΓE(R), the relationship of metric dimension between ΓE(R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of ΓE(R). We provide a formula for the number of vertices of the family of graphs given by ΓE(R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of ΓE(R).

Computing metric dimension of compressed zero divisor graphs associated to rings

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"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 ΓE(R) with vertex set Z(RE) \ {[0]} = RE \ {[0], [1]}...

Computing metric dimension of compressed zero divisor graphs associated to rings

Published Online: Mar 04, 2019

Page range: 298 - 318

Received: Aug 06, 2018

DOI: https://doi.org/10.2478/ausm-2018-0023

Keywords
metric dimension, zero-divisor graph, compressed zero-divisor graph, equivalence classes

© 2018 S. Pirzada et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Computing metric dimension of compressed zero divisor graphs associated to rings

Published Online: Mar 04, 2019

Page range: 298 - 318

Received: Aug 06, 2018

DOI: https://doi.org/10.2478/ausm-2018-0023

Keywordsmetric dimension, zero-divisor graph, compressed zero-divisor graph, equivalence classes

© 2018 S. Pirzada et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
metric dimension, zero-divisor graph, compressed zero-divisor graph, equivalence classes