On the metric dimension of barycentric subdivision of Cayley graphs

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6 Citations (Scopus)

Abstract

In a connected graph G, the distance d(u, v) denotes the distance between two vertices u and v of G. Let W = {w1, w2, ···, wk} be an ordered set of vertices of G and let v be a vertex of G. The representationr(v|W) of v with respect to W is the k-tuple (d(v, w1), d(v,w2), ···, d(v, wk)). 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 (Zn ⨁ Z2). 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 (Zn ⨁ Z2).

Original languageEnglish
Pages (from-to)1067-1072
Number of pages6
JournalActa Mathematicae Applicatae Sinica
Volume32
Issue number4
DOIs
Publication statusPublished - Oct 1 2016
Externally publishedYes

Keywords

  • Cayley graph
  • barycentric subdivision
  • basis
  • metric dimension
  • resolving set

ASJC Scopus subject areas

  • Applied Mathematics

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