# Upper Dimension and Bases of Zero-divisor Graphs of Commutative Rings

## Department

Mathematics and Statistics

## Document Type

Article

## Publication Date

2020

## Abstract

Summary: ``For a commutative ring $R$ with non-zero zero divisor set $Z\sp*(R)$, the zero divisor graph of $R$ is $\Gamma(R)$ with vertex set $Z\sp*(R)$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$. The upper dimension and the resolving number of a zero divisor graph $\Gamma(R)$ of some rings are determined. We provide certain classes of rings which have the same upper dimension and metric dimension and give an example of a ring for which these values do not coincide. Further, we obtain some bounds for the upper dimension in zero divisor graphs of commutative rings and provide a subset of vertices which cannot be excluded from any resolving set.''

## Recommended Citation

Pirzada, S., Aijaz, M., & Redmond, S. P. (2020). Upper Dimension and Bases of Zero-divisor Graphs of Commutative Rings. AKCE International Journal of Graphs and Combinatorics, 17(1), 168.

## Journal Title

AKCE International Journal of Graphs and Combinatorics