Adjacency metric dimension of the 2-absorbing ideals graph
Subject Areas : StatisticsS. B. Pejman 1 , Sh. Payrovi 2 , A. Behtoei 3
1 - Department of Mathematics, Imam Khomeini International University
2 - Department of Mathematics, Imam Khomeini International University
3 - Department of Mathematics, Imam Khomeini International University
Keywords: ایدهآل دوجاذب, شمارنده صفر, گراف ردههای همارزی, بعد متری مجاورتی, مجموعه کاشف,
Abstract :
Let Γ=(V,E) be a graph and W_(a)=\{w_1,…,w_k \} be a subset of the vertices of Γ and v be a vertex of it. The k-vector r_2 (v∣ W_a)=(a_Γ (v,w_1),… ,a_Γ (v,w_k)) is the adjacency representation of v with respect to W in which a_Γ (v,w_i )=min\{2,d_Γ (v,w_i )\} and d_Γ (v,w_i ) is the distance between v and w_i in Γ. W_a is called as an adjacency resolving set for Γ if distinct vertices of Γ have distinct adjacency representations w.r.t W_a. The size of the smallest adjacency resolving set is the adjacency metric dimension of Γ and is denoted by dim_a(Γ). In this paper, we prove that dim_a(Γ_E (Z_(P^n ) ))=⌈(n-2)/2⌉. Also, we show that Γ_E (Z_(p^2n ) )≅Γ_E (R/I) in which p is a prime number, n is a natural number and I is a 2-absorbing ideal of the ring R which has a minimal primitive decomposition in the form of the intersection of n primitive ideals. Finally we conclude that dim_a〖(Γ_E (R/I))=n-1〗.
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