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On strong metric dimension of zero-divisor graphs of rings. (English) Zbl 1427.13003

Summary: In this paper, we study the strong metric dimension of zero-divisor graph \(\Gamma(R)\) associated to a ring \(R\). This is done by transforming the problem into a more well-known problem of finding the vertex cover number \(\alpha(G)\) of a strong resolving graph \(G_{sr}\). We find the strong metric dimension of zero-divisor graphs of the ring \(\mathbb{Z}_n\) of integers modulo \(n\) and the ring of Gaussian integers \(\mathbb{Z}_n[i]\) modulo \(n\). We obtain the bounds for strong metric dimension of zero-divisor graphs and we also discuss the strong metric dimension of the Cartesian product of graphs.

MSC:

13A05 Divisibility and factorizations in commutative rings
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C12 Distance in graphs
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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