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Communicated by S. R. Lopez-PermouthFor a graph G(V, E) with order n ≥ 2, the locating code of a vertex v is a finite vector representing distances of v with respect to vertices of some ordered subset W of V (G). The set W is a locating set of G(V, E) if distinct vertices have distinct codes. A locating set containing a minimum number of vertices is a minimum locating set for G(V, E). The locating number denoted by loc(G) is the number of vertices in the minimum locating set. Let R be a commutative ring with identity 1 = 0, the zero-divisor graph denoted by Γ(R), is the (undirected) graph whose vertices are the nonzero zero-divisors of R with two distinct vertices joined by an edge when the product of vertices is zero. We introduce and investigate locating numbers in zero-divisor graphs of a commutative ring R. We then extend our definition to study and characterize the locating numbers of an ideal based zero-divisor graph of a commutative ring R.

Abstract. Let M be an R-module, where R is a commutative ring with identity 1 and let G(V, E) be a graph. In this paper, we study the graphs associated with modules over commutative rings. We associate three simple graphs ann f (Γ(M R )), anns(Γ(M R )) and annt(Γ(M R )) to M called full annihilating, semi-annihilating and star-annihilating graph. When M is finite over R, we investigate metric dimensions in ann f (Γ(M R )), anns(Γ(M R )) and annt(Γ(M R )). We show that M over R is finite if and only if the metric dimension of the graph ann f (Γ(M R )) is finite. We further show that the graphs ann f (Γ(M R )), anns(Γ(M R )) and annt(Γ(M R )) are empty if and only if M is a prime-multiplicationlike R-module. We investigate the case when M is a free R-module, where R is an integral domain and show that the graphs ann f (Γ(M R )), anns(Γ(M R )) and annt(Γ(M R )) are empty if and only if M ∼ = R. Finally, we characterize all the non-simple weakly virtually divisible modules M for which Ann(M ) is a prime ideal and Soc(M ) = 0.

Let R be a commutative ring with identity and let G(V, E) be a graph. The locating number of the graph G(V, E) denoted by loc (G) is the cardinality of the minimal locating set W ⊆ V(G). To get the loc (G), we assign locating codes to the vertices V(G)∖W of G in such a way that every two vertices get different codes. In this paper, we consider the ratio of loc (G) to |V(G)| and show that there is a finite connected graph G with loc (G)/|V(G)| = m/n, where m < n are positive integers. We examine two equivalence relations on the vertices of Γ(R) and the relationship between locating sets and the cut vertices of Γ(R). Further, we obtain bounds for the locating number in zero-divisor graphs of a commutative ring and discuss the relation between locating number, domination number, clique number and chromatic number of Γ(R). We also investigate the locating number in Γ(R) when R is a finite product of rings.

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