No 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.

No abstract

Let R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that {\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}}, n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.

The metric representation of a vertex [Formula: see text] of a graph [Formula: see text] is a finite vector representing distances of [Formula: see text] with respect to vertices of some ordered subset [Formula: see text]. The set [Formula: see text] is called a minimal resolving set if no proper subset of [Formula: see text] gives distinct representations for all vertices of [Formula: see text]. The metric dimension of [Formula: see text] is the cardinality of the smallest (with respect to its cardinality) minimal resolving set and upper dimension is the cardinality of the largest minimal resolving set. We show the existence of graphs for which metric dimension equals upper dimension. We found an error in a result, defining the metric dimension of join of path and totally disconnected graph, of the paper by Shahida and Sunitha [On the metric dimension of join of a graph with empty graph ([Formula: see text]), Electron. Notes Discrete Math. 63 (2017) 435–445] and we give the correct form of the theorem and its proof.