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On the metric dimension of barycentric subdivision of Cayley graphs

On the metric dimension of barycentric subdivision of Cayley graphs In a connected graph G, the distance d(u, v) denotes the distance between two vertices u and v of G. Let W = {w 1, w 2, ···, w k} be an ordered set of vertices of G and let v be a vertex of G. The representation r(v|W) of v with respect to W is the k-tuple (d(v, w 1), d(v,w 2), ···, d(v, w k)). The set W is called a resolving set or a locating set if every vertex of G is uniquely identified by its distances from the vertices of W, or equivalently, if distinct vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for G and this cardinality is the metric dimension of G, denoted by β(G). Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). In this paper, we study the metric dimension of barycentric subdivision of Cayley graphs Cay (Z n ⨁ Z 2). We prove that these subdivisions of Cayley graphs have constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of barycentric subdivision of Cayley graphs Cay (Z n ⨁ Z 2). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

On the metric dimension of barycentric subdivision of Cayley graphs

Imran, Muhammad
Acta Mathematicae Applicatae Sinica , Volume 32 (4) – Oct 1, 2016
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References (23)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-016-0627-0
Publisher site
See Article on Publisher Site

Abstract

In a connected graph G, the distance d(u, v) denotes the distance between two vertices u and v of G. Let W = {w 1, w 2, ···, w k} be an ordered set of vertices of G and let v be a vertex of G. The representation r(v|W) of v with respect to W is the k-tuple (d(v, w 1), d(v,w 2), ···, d(v, w k)). The set W is called a resolving set or a locating set if every vertex of G is uniquely identified by its distances from the vertices of W, or equivalently, if distinct vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for G and this cardinality is the metric dimension of G, denoted by β(G). Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). In this paper, we study the metric dimension of barycentric subdivision of Cayley graphs Cay (Z n ⨁ Z 2). We prove that these subdivisions of Cayley graphs have constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of barycentric subdivision of Cayley graphs Cay (Z n ⨁ Z 2).

Journal

Acta Mathematicae Applicatae Sinica – Springer Journals

Published: Oct 1, 2016

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