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〗.
[1] N. Biggs, Algebraic Graph Theory, Cambridge University Press, 1974.
[2] J. Caceres, C. Hernando, M. Mora, I.M. Pelayo, M.L. Puertas, C. Seara and D.R. Wood, On the metric dimension of some families of graphs, Electron. Notes Discrete Math., 22(2005), pp. 129-133.
[3] J. Caceres, C. Hernando, M. Mora, I.M. Pelayo, M.L. Puertas, C. Seara and D.R. Wood, On the metric dimension of cartesian product of graphs, SIAM, J. Disc. Math., 2 (2007), pp. 423-441.
[4] G. Chartrand, L. Eroh, M.A. Johnson, and O.R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math., 105(2000), pp. 99-113.
[5] F. Harary, R.A. Melter, On the Metric Dimension of a graph, Ars Combin., 2(1976), pp. 191-195.
[6] C. Hernando, M. Mora, I.M. Pelayo, C. Seara, D. R. Wood, Extremal Graph Theory for Metric Dimension and Diameter, Electron. J. Combin., 17(2010), pp. 1-28.
[7] M. Jannesari, B. Omoomi, The metric dimension of the lexicographic product of graphs, Discrete Math., 312(2012), pp. 3349-3356.
[8] M. Jannesari, B. Omoomi, Characterization of n-vertex graphs with metric dimension n-3, Math. Bohem., 139(2014), pp. 1-23.
[9] Sh. Payrovi, S. Babaei, On the 2-absorbing ideals and zero divisor graph of equivalence classes of zero divisors, Journal of Hyperstructures, 3 (2014), pp. 1-9.
[10] R.Y. Sharp, Steps in Commutative Algebra, second edition, London Mathematical Society Student Texts 51, Cambridge University Press, Cambridge, 2000.
[11] P.J. Slater, Leaves of trees, Congressus Numerantium 14(1975) pp. 549-559.
[12] S. Spiroff, C. Wickham, A zero divisor graph determind by equivalence classes of zero divisors, Comm. Algebra, 39(2011), pp. 2338-2348.
[13] F. Levidiotis , S. Spiroff, Five-point zero-divisor graphs determined by equivalence classes ,Involve, 4 (2011), pp. 53-64.
