<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

Publicado en línea: 04 mar 2019

Páginas: 298 - 318

Recibido: 06 ago 2018

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

Palabras clave
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

Publicado en línea: 04 mar 2019

Páginas: 298 - 318

Recibido: 06 ago 2018

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

Palabras clavemetric 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.

Palabras clave
metric dimension, zero-divisor graph, compressed zero-divisor graph, equivalence classes