Open Mathematics (Nov 2019)

Power graphs and exchange property for resolving sets

  • Abbas Ghulam,
  • Ali Usman,
  • Munir Mobeen,
  • Bokhary Syed Ahtsham Ul Haq,
  • Kang Shin Min

DOI
https://doi.org/10.1515/math-2019-0093
Journal volume & issue
Vol. 17, no. 1
pp. 1303 – 1309

Abstract

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Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

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