Upper dimension and bases of zero-divisor graphs of commutative rings. Issue 1 (2nd January 2020)

• Record Type: Journal Article
• Authors: Pirzada, S.; Aijaz, M.; Redmond, S.P.
• Is Part Of: AKCE International Journal of Graphs and Combinatorics. Volume 17:Issue 1(2020)
• Page Start: 168
• Page End: 173
• Publication Date: 2020-01-02
• Subjects: Ring -- Zero-divisor graph -- Upper dimension -- Resolving number
• DOI: 10.1016/j.akcej.2018.12.001
• Languages: English
• ISSNs: 0972-8600

Abstract

Abstract: For a commutative ring R with non-zero zero divisor set Z ∗ ( R ), the zero divisor graph of R is Γ ( R ) with vertex set Z ∗ ( R ), where two distinct vertices x and y are adjacent if and only if x y = 0 . The upper dimension and the resolving number of a zero divisor graph Γ ( 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.