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Symmetry 2018, 10(6), 209; https://doi.org/10.3390/sym10060209

Computing the Metric Dimension of Gear Graphs

1
Department of Mathematics, COMSATS University Islamabad, Lahore Campus 54000, Pakistan
2
Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus 57000, Pakistan
3
Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, United Arab Emirates
4
Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad 44000, Pakistan
5
Department of Electrical Engineering, University of Central Punjab, Lahore 54000, Pakistan
6
Department of Mathematics, The University of Lahore, Old Campus Lahore 54000, Pakistan
*
Author to whom correspondence should be addressed.
Received: 15 May 2018 / Revised: 5 June 2018 / Accepted: 6 June 2018 / Published: 8 June 2018
(This article belongs to the Special Issue Symmetry in Graph Theory)
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Abstract

Let G = (V, E) be a connected graph and d(u, v) denote the distance between the vertices u and v 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). Let J2n,m be a m-level gear graph obtained by m-level wheel graph W2n,mmC2n + k1 by alternatively deleting n spokes of each copy of C2n and J3n be a generalized gear graph obtained by alternately deleting 2n spokes of the wheel graph W3n. In this paper, the metric dimension of certain gear graphs J2n,m and J3n generated by wheel has been computed. Also this study extends the previous result given by Tomescu et al. in 2007. View Full-Text
Keywords: Metric dimension; basis; resolving set; gear graph; generalized gear graph Metric dimension; basis; resolving set; gear graph; generalized gear graph
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Imran, S.; Siddiqui, M.K.; Imran, M.; Hussain, M.; Bilal, H.M.; Cheema, I.Z.; Tabraiz, A.; Saleem, Z. Computing the Metric Dimension of Gear Graphs. Symmetry 2018, 10, 209.

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