On determining number and metric dimension of zero-divisor graphs

TL;DR

This paper provides explicit formulas and bounds for the determining number and metric dimension of zero-divisor graphs of Z_n and semisimple rings, and addresses an open problem in graph theory.

Contribution

It introduces formulas for zero-divisor graphs and resolves an open problem on the determining number of graphs.

Findings

Formulas for zero-divisor graphs of Z_n and semisimple rings.

Upper bounds for Boolean rings' zero-divisor graphs.

Resolution of Boutin's open problem on graph determining numbers.

Abstract

In this article, explicit formulas for finding the determining number and the metric dimension of the zero-divisor graph of Z_n and non-Boolean semisimple rings are given. In the case of Boolean rings, an upper bound of the determining number and the metric dimension of zero-divisor graph is determined. Further, the determining number and the metric dimension of some important graphs other than zero-divisor graphs, are proved and the open problem by Boutin, regarding the determining number of graphs is settled.