A set W ⊆ V (G) is called a resolving set, if for each pair of distinct verticesis the distance between vertices x and y. The cardinality of a minimum resolving set for G is called the metric dimension of G and is denoted by dim M (G). This parameter has many applications in different areas. The problem of finding metric dimension is NPcomplete for general graphs but it is determined for trees and some other important families of graphs. In this paper, we determine the exact value of the metric dimension of Andrásf ai graphs, their complements and And(k) P n . Also, we provide upper and lower bounds for dim M (And(k) C n ).

Let R be a commutative ring and M a Noetherian R-module. The zero-divisor graph of M , denoted by Γ(M), is an undirected simple graph whose vertices are the elements of Z R (M) Ann R (M) and two distinct vertices a and b are adjacent if and only if abM = 0. In this paper, we study diameter and girth of Γ(M). We show that the zero-divisor graph of M has a universal vertex in Z R (M) r(Ann R (M)) if and only if R = ⊕Z 2 ⊕R ′ and M = Z 2 ⊕M ′ , where M ′ is an R ′-module. Moreover, we show that if Γ(M) is a complete graph, then one of the following statements is true: (i) Ass R (M) = {m 1 , m 2 }, where m 1 , m 2 are maximal ideals of R. (ii) Ass R (M) = {p}, where p 2 ⊆ Ann R (M). (iii) Ass R (M) = {p}, where p 3 ⊆ Ann R (M).

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Let R be a commutative Noetherian ring, I, J two proper ideals of R and let M be a non-zero finitely generated R-module with c = cd(I, J, M). In this paper, we first introduce T R (I, J, M) as the largest submodule of M with the property that cd(I, J, T R (I, J, M)) < c and we describe it in terms of the reduced primary decomposition of zero submodule of M. It is shown that Ann R (H d I,J (M)) = Ann R (M/T R (I, J, M)) and Ann R (H d I (M)) = Ann R (H d I,J (M)), whenever R is a local ring, M has dimension d with H d I,J (M) = 0 and J t M ⊆ T R (I, M) for some positive integer t.

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