Characterization of n -Vertex Graphs of Metric Dimension n ? 3 by Metric Matrix

Juan Wang, Lianying Miao and Yunlong Liu
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Juan Wang: School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
Lianying Miao: School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
Yunlong Liu: College of Information and Control Engineering, Weifang University, Weifang 261061, China

Mathematics, 2019, vol. 7, issue 5, 1-13

Abstract: Let G = ( V ( G ) , E ( G ) ) be a connected graph. An ordered set W ⊂ V ( G ) is a resolving set for G if every vertex of G is uniquely determined by its vector of distances to the vertices in W . The metric dimension of G is the minimum cardinality of a resolving set. In this paper, we characterize the graphs of metric dimension n − 3 by constructing a special distance matrix, called metric matrix. The metric matrix makes it so a class of graph and its twin graph are bijective and the class of graph is obtained from its twin graph, so it provides a basis for the extension of graphs with respect to metric dimension. Further, the metric matrix gives a new idea of the characterization of extremal graphs based on metric dimension.

Keywords: extremal graph; metric dimension; resolving set; metric matrix (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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