TY - JOUR

T1 - On the metric dimension of barycentric subdivision of Cayley graphs

AU - Imran, Muhammad

N1 - Funding Information:
Supported by the National University of Sciences and Technology (NUST), H-12, Islamabad, Pakistan.
Publisher Copyright:
© 2016, Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg.

PY - 2016/10/1

Y1 - 2016/10/1

N2 - 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).

AB - 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).

KW - Cayley graph

KW - barycentric subdivision

KW - basis

KW - metric dimension

KW - resolving set

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U2 - 10.1007/s10255-016-0627-0

DO - 10.1007/s10255-016-0627-0

M3 - Article

AN - SCOPUS:84989355118

SN - 0168-9673

VL - 32

SP - 1067

EP - 1072

JO - Acta Mathematicae Applicatae Sinica

JF - Acta Mathematicae Applicatae Sinica

IS - 4

ER -