Computing the metric dimension of gear graphs

Shahid Imran; Muhammad Kamran Siddiqui; Muhammad Imran; Muhammad Hussain; Hafiz Muhammad Bilal; Imran Zulfiqar Cheema; Ali Tabraiz; Zeeshan Saleem

doi:10.3390/sym10060209

Computing the metric dimension of gear graphs

Shahid Imran, Muhammad Kamran Siddiqui, Muhammad Imran, Muhammad Hussain, Hafiz Muhammad Bilal, Imran Zulfiqar Cheema, Ali Tabraiz, Zeeshan Saleem

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

14 Citations (Scopus)

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 J_2n,m be a m-level gear graph obtained by m-level wheel graph W_2n,m ≅ mC_2n + k₁ by alternatively deleting n spokes of each copy of C_2n and J_3n be a generalized gear graph obtained by alternately deleting 2n spokes of the wheel graph W_3n. In this paper, the metric dimension of certain gear graphs J_2n,m and J_3n generated by wheel has been computed. Also this study extends the previous result given by Tomescu et al. in 2007.

Original language	English
Article number	209
Journal	Symmetry
Volume	10
Issue number	6
DOIs	https://doi.org/10.3390/sym10060209
Publication status	Published - Jun 1 2018

Keywords

Basis
Gear graph
Generalized gear graph
Metric dimension
Resolving set

ASJC Scopus subject areas

Computer Science (miscellaneous)
Chemistry (miscellaneous)
Mathematics(all)
Physics and Astronomy (miscellaneous)

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10.3390/sym10060209

Cite this

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title = "Computing the metric dimension of gear graphs",

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,m ≅ mC2n + 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.",

keywords = "Basis, Gear graph, Generalized gear graph, Metric dimension, Resolving set",

author = "Shahid Imran and Siddiqui, {Muhammad Kamran} and Muhammad Imran and Muhammad Hussain and Bilal, {Hafiz Muhammad} and Cheema, {Imran Zulfiqar} and Ali Tabraiz and Zeeshan Saleem",

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AU - Imran, Shahid

AU - Siddiqui, Muhammad Kamran

AU - Imran, Muhammad

AU - Hussain, Muhammad

AU - Bilal, Hafiz Muhammad

AU - Cheema, Imran Zulfiqar

AU - Tabraiz, Ali

AU - Saleem, Zeeshan

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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,m ≅ mC2n + 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.

AB - 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,m ≅ mC2n + 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.

KW - Basis

KW - Gear graph

KW - Generalized gear graph

KW - Metric dimension

KW - Resolving set

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Computing the metric dimension of gear graphs

Abstract

Keywords

ASJC Scopus subject areas

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