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On the Metric Dimension of barycentric subdivision of Cayley Graphs

Acta Mathematicae Applicatae Sinica 2016年第4期32卷 1067-1072页

作者： Muhammad IMRAN Department of Mathematics School of Natural Sciences National University of Sciences and Technology Sector H-12 Islamabad Pakistan

In a connected graph G, the distance d（u, v） denotes the distance between two vertices u and v of G. Let W = {w1, w2,……, wk} be an ordered set of vertices of G and let v be a vertex of G. The representation r（v1W） of v with respect to W is the k-tuple （d（v, w1）, d（v, w2）,…, d（v, wk））. The set W is called a resolving set or a locating set if every vertex of G is uniquely identified by its distances from the vertices of W, or equivalently, if distinct vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for G and this cardinality is the metric dimension of G, denoted by β（G）. Metric dimension is a generalization of affine dimension to arbitrary metric spaces （provided a resolving set exists）. In this paper, we study the metric dimension of barycentric subdivision of Cayley graphs Cay （Zn Z2）. We prove that these subdivisions of Cayley graphs have constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of barycentric subdivision of Cayley graphs Cay （Zn Z2）.

关键词： metric dimension basis resolving set barycentric subdivision Cayley graph

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Some geometric properties of successive difference substitutions

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Science China(Information Sciences) 2011年第4期54卷 778-786页

作者： HOU XiaoRong 1,XU Song 1,2 & SHAO JunWei 1 1 School of Automation Engineering,University of Electronic Science and Technology of China,Chengdu 611731,China 2 Faculty of Science,Ningbo University,Ningbo 315211,China 1. School of Automation Engineering University of Electronic Science and Technology of China Chengdu 611731 China2. Faculty of Science Ningbo University Ningbo 315211 China

This paper provides a new,geometric perspective to study successive difference substitutions,and proves that the sequence of the successive difference substitution sets is not *** interesting result that a given k-dimensional rational hyperplane can be transformed to a k-dimensional coordinate hyperplane of new variables by finite steps of successive difference substitutions is ***,a sufficient condition for the sequence of the successive difference substitution sets of a form being not terminating is *** is,a class of polynomials which cannot be proved to be positive semi-definite by the successive difference substitution method are obtained.

关键词： successive difference substitutions nonnegativity decision of forms barycentric subdivision

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