On the metric dimension of a zero-divisor graph. Issue 4 (3rd April 2017)

• Record Type: Journal Article
• Authors: Pirzada, S.; Raja, Rameez
• Is Part Of: Communications in algebra. Volume 45:Issue 4(2017)
• Page Start: 1399
• Page End: 1408
• Publication Date: 2017-04-03
• Subjects: Metric dimension -- ring -- zero-divisor -- zero-divisor graph; 13A99 -- 05C78 -- 05C12; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2016.1175602
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: Let R be a commutative ring with unity 1 and let G ( V, E ) be a simple graph. In this research article, we study the metric dimension in zero-divisor graphs associated with commutative rings. We show that for a given rational q ∈(0, 1), there exists a finite graph G such that the ratio d i m M ( G ) | V ( H ) | = q, where H is any induced connected subgraph of G . We provide a metric dimension formula for a zero-divisor graph Γ ( R × 𝔽 q ) and give metric dimension of the zero-divisor graph Γ ( R 1 × R 2 × ⋯ × R n ), where R 1, R 2, …, R n are n finite commutative rings with each having unity 1 and none of R i, 1≤ i ≤ n, being isomorphic to the Boolean ring ∏ i = 1 n ℤ 2 . We discuss the metric dimension of Cartesian product of zero-divisor graphs and show that there exists a zero-divisor graph Γ ( R 1 ) × Γ ( R 2 ) such that d i m M ( Γ ( R 1 ) × Γ ( R 2 ) ) lies between the numbers d i m M ( Γ ( R 1 ) ) and d i m M ( Γ ( R 1 ) ) + 1, where R 1 and R 2 are any two finite commutative rings (not domains) with each having unity 1.