Pirzada, S., Aijaz, M., & Redmond, S. (2019). Upper dimension and bases of zero-divisor graphs of commutative rings. AKCE International Journal of Graphs and Combinatorics. doi:10.1016/j.akcej.2018.12.001
Mathematics and Statistics
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.
AKCE International Journal of Graphs and Combinatorics