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On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

Author

Listed:
  • Shahid Imran

    (Govt Khawaja Rafique Shaheed College Walton Road Lahore, Lahore 54000, Pakistan)

  • Muhammad Kamran Siddiqui

    (Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus, Punjab 57000, Pakistan
    Department of Mathematical Sciences, United Arab Emirates University, Al Ain, P.O. Box 15551, UAE)

  • Muhammad Imran

    (Department of Mathematical Sciences, United Arab Emirates University, Al Ain, P.O. Box 15551, UAE
    Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad 44000, Pakistan)

  • Muhammad Hussain

    (Department of Mathematics, COMSATS University Islamabad, Lahore Campus 54000, Pakistan)

Abstract

Let G = ( V , E ) be a connected graph and d ( x , y ) be the distance between the vertices x and y in G . A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W . A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim ( G ). In this paper, Cycle, Path, Harary graphs and their rooted product as well as their connectivity are studied and their metric dimension is calculated. It is proven that metric dimension of some graphs is unbounded while the other graphs are constant, having three or four dimensions in certain cases.

Suggested Citation

  • Shahid Imran & Muhammad Kamran Siddiqui & Muhammad Imran & Muhammad Hussain, 2018. "On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product," Mathematics, MDPI, vol. 6(10), pages 1-16, October.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:10:p:191-:d:174087
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    References listed on IDEAS

    as
    1. András Sebő & Eric Tannier, 2004. "On Metric Generators of Graphs," Mathematics of Operations Research, INFORMS, vol. 29(2), pages 383-393, May.
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