<abstract xmlns="http://www.w3.org/1999/xhtml">Let R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausi-2020-0006_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtext>R</mtext><mo>≅</mo><mi>Π</mi><msubsup><mrow><mtext>𝕑</mtext></mrow><mn>2</mn><mtext>n</mtext></msubsup></mrow></math><tex-math>{\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}}</tex-math></alternatives></inline-formula>, n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.</abstract>

Let R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that R≅Π𝕑2n{\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}}, n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.

Metric and upper dimension of zero divisor graphs associated to commutative rings

Acta Universitatis Sapientiae, Informatica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Let R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is...

Metric and upper dimension of zero divisor graphs associated to commutative rings

Published Online: Jul 16, 2020

Page range: 84 - 101

Received: Feb 20, 2020

Accepted: Mar 23, 2020

DOI: https://doi.org/10.2478/ausi-2020-0006

Keywords
ring, zero divisor, zero divisor graph, metric dimension, upper dimension

© 2020 S. Pirzada et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Metric and upper dimension of zero divisor graphs associated to commutative rings

Published Online: Jul 16, 2020

Page range: 84 - 101

Received: Feb 20, 2020

Accepted: Mar 23, 2020

DOI: https://doi.org/10.2478/ausi-2020-0006

Keywordsring, zero divisor, zero divisor graph, metric dimension, upper dimension

© 2020 S. Pirzada et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
ring, zero divisor, zero divisor graph, metric dimension, upper dimension