Metric and upper dimension of zero divisor graphs associated to commutative rings

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.

Co-unit graphs associated to ring of integers modulo n

“…We investigated some basic properties of G nu (R), where R is the ring of integers modulo n, for different values of n. We obtained the domination number, the clique number and the girth of G nu (R). For the future work, we need to investigate several other graph invariants of G nu (R), for any ring R. Also, there is scope to study the line graph of the co-unit graph, in analogy to the line graph of the unit graph, see [10]. Further directions to study in co-unit graphs can be metric dimension and spectra, for instance like in [3,9,10,11,12].…”

“…For the future work, we need to investigate several other graph invariants of G nu (R), for any ring R. Also, there is scope to study the line graph of the co-unit graph, in analogy to the line graph of the unit graph, see [10]. Further directions to study in co-unit graphs can be metric dimension and spectra, for instance like in [3,9,10,11,12].…”

A graph-theoretic approach to ring analysis: Dominant metric dimensions in zero-divisor graphs

“…In 2019 [ 28 ], the metric dimension of ZD-graphs for ring was calculated. In 2020 [ 29 ], bounds for the EMD of ZD-graphs related to rings were studied by Siddiqui et al Pirzada and Aijaz in 2020 [ 30 ] studied ZD-graphs for commutative rings for their metric and upper dimension.…”

On the Strong Metric Dimension of a Total Graph of Nonzero Annihilating Ideals

“…A number of results have been presented on the strong metric dimension of cartesian product graphs and Cayley graphs [10] and distance-hereditary graphs [9]. Later, the metric dimension and strong metric dimension were applied to graphs associated to commutative rings (see, for example, [4,5,[11][12][13]). In [1], the authors studied the metric dimension of the total graph of nonzero annihilating ideals.…”