On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

Mathematics

Muhammad Imran¹  Muhammad Hussain²  MuhammadKamran Siddiqui³  Shahid Imran⁴ 

[1]Department of Mathematical Sciences, United Arab Emirates University, Al Ain, P.O. Box 15551, UAE

[2]Department of Mathematics, COMSATS University Islamabad, Lahore Campus 54000, Pakistan

[3]Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus, Punjab 57000, Pakistan

[4]Govt Khawaja Rafique Shaheed College Walton Road Lahore, Lahore 54000, Pakistan

关键词: metric dimension;    basis;    resolving set;    cycle;    path;    Harary graphs;    rooted product;   

DOI  :  10.3390/math6100191

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown