On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

Shahid Imran, Muhammad Kamran Siddiqui, Muhammad Imran and Muhammad Hussain
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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
Muhammad Imran: Department of Mathematical Sciences, United Arab Emirates University, Al Ain, P.O. Box 15551, UAE
Muhammad Hussain: Department of Mathematics, COMSATS University Islamabad, Lahore Campus 54000, Pakistan

Mathematics, 2018, vol. 6, issue 10, 1-16

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.

Keywords: metric dimension; basis; resolving set; cycle; path; Harary graphs; rooted product (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2018
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