On the metric dimension of circulant graphs

Imran, Muhammad; Baig, Abdul Qudair; Bokhary, Syed Ahtsham Ul Haq; Javaid, Imran

doi:10.1016/j.aml.2011.09.008

Cited by 60 publications

(36 citation statements)

References 9 publications

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“…A resolving set of minimum cardinality is called a basis for G and this cardinality is the metric dimension or location number of G, denoted by β (G) [5]. The concepts of resolving set and metric basis have previously appeared in the literature (see [1][2][3][4][5][6][7][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]). …”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Metric Dimension and Exchange Property for Resolving Sets in Rotationally-Symmetric Graphs

Naeem¹,

Imran²

2014

Appl. Math. Inf. Sci.

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Abstract:Metric dimension or location number is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let F be a family of connected graphs G n : F = (G n ) n≥1 depending on n as follows: the order |V (G)| = ϕ(n) and lim n→∞ ϕ(n) = ∞. If there exists a constant C > 0 such that dim(G n ) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension, otherwise F has unbounded metric dimension. If all graphs in F have the same metric dimension (which does not depend on n), F is called a family with constant metric dimension.In this paper, we study the metric dimension of quasi flower snarks, generalized antiprism and cartesian product of square cycle and path. We prove that these classes of graphs have constant or bounded metric dimension. It is natural to ask for characterization of graphs classes with respect to the nature of their metric dimension. It is also shown that the exchange property of the bases in a vector space does not hold for minimal resolving sets of quasi flower snarks, generalized prism and generalized antiprism.

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Metric Dimension and Exchange Property for Resolving Sets in Rotationally-Symmetric Graphs

Naeem¹,

Imran²

2014

Appl. Math. Inf. Sci.

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“…Javaid et al [9] initiated a study of the metric dimension of circulants as some classes of these graphs had been shown to have bounded metric dimension. Imran et al [8] later bounded the metric dimension of C n (1, 2) and C n (1, 2, 3), and then Borchert and Gosselin [2] extended their results and determined the exact metric dimension of these two families of circulants for all n.…”

Section: History and Layout Of The Papermentioning

confidence: 99%

The metric dimension of circulant graphs and their Cartesian products

Chau

Gosselin

2017

OpMath

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Abstract. Let G = (V, E) be a connected graph (or hypergraph) and let d(x, y) denote the distance between verticesThe minimum cardinality of a resolving set for G is called the metric dimension of G, denoted by β(G). The circulant graph Cn(1, 2, . . . , t) has vertex set {v0, v1, . . . , vn−1} and edges vivi+j where 0 ≤ i ≤ n − 1 and 1 ≤ j ≤ t and the indices are taken modulo n (2 ≤ t ≤ n 2 ). In this paper we determine the exact metric dimension of the circulant graphs Cn(1, 2, . . . , t), extending previous results due to Borchert and Gosselin (2013), Grigorious et al. (2014), andVetrík (2016). In particular, we show that β (Cn(1, 2, . . . , t)) = β (Cn+2t(1, 2, . . . , t)) for large enough n, which implies that the metric dimension of these circulants is completely determined by the congruence class of n modulo 2t. We determine the exact value of β(Cn (1, 2, . . . , t)) for n ≡ 2 mod 2t and n ≡ (t + 1) mod 2t and we give better bounds on the metric dimension of these circulants for n ≡ 0 mod 2t and n ≡ 1 mod 2t. In addition, we bound the metric dimension of Cartesian products of circulant graphs.

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“…computed the metric dimension of a honey-comb network in [21]. In [22], the authors computed the metric dimension of circulant graphs. In [23], authors computed explicit formula for the metric dimension of a regular bipartite graph.…”

Section: Introductionmentioning

confidence: 99%

Computing Metric Dimension and Metric Basis of 2D Lattice of Alpha-Boron Nanotubes

et al. 2018

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Concepts of resolving set and metric basis has enjoyed a lot of success because of multi-purpose applications both in computer and mathematical sciences. For a connected graph G(V,E) a subset W of V(G) is a resolving set for G if every two vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for graph G and this minimum cardinality is known as metric dimension of G. Boron nanotubes with different lattice structures, radii and chirality's have attracted attention due to their transport properties, electronic structure and structural stability. In the present article, we compute the metric dimension and metric basis of 2D lattices of alpha-boron nanotubes.

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On the metric dimension of circulant graphs

Cited by 60 publications

References 9 publications

Metric Dimension and Exchange Property for Resolving Sets in Rotationally-Symmetric Graphs

Metric Dimension and Exchange Property for Resolving Sets in Rotationally-Symmetric Graphs

The metric dimension of circulant graphs and their Cartesian products

Computing Metric Dimension and Metric Basis of 2D Lattice of Alpha-Boron Nanotubes

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