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Metric Basis of Four-Dimensional Klein Bottle

Computer Modeling in Engineering & Sciences 2023年第9期136卷 3011-3024页

作者： Ali N.A.Koam Ali Ahmad Maryam Salem Alatawi Muhammad Azeem Muhammad Faisal Nadeem Department of Mathematics College of SciencesNew CampusJazan UniversityJazanSaudi Arabia College of Computer Science&Information Technology Jazan UniversityJazanSaudi Arabia Department of Mathematics Faculty of SciencesUniversity of TabukTabukSaudi Arabia Department of Mathematics Riphah Institute of Computing and Applied SciencesRiphah International UniversityLahorePakistan Department of Mathematics COMSATS University IslamabadLahore CampusLahorePakistan

The Metric of a graph plays an essential role in the arrangement of different dimensional structures and finding their basis in various *** metric dimension of a graph is the selection of the minimum possible number of vertices so that each vertex of the graph is distinctively defined by its vector of distances to the set of selected *** set of selected vertices is known as the metric basis of a *** applied mathematics or computer science,the topic of metric basis is considered as locating number or locating set,and it has applications in robot navigation and finding a beacon set of a computer *** to the vast applications of this concept in computer science,optimization problems,and also in chemistry enormous research has been *** extend this research to a four-dimensional structure,we studied the metric basis of the Klein bottle and proved that the Klein bottle has a constant metric dimension for the variation of all its *** the metric basis is variying in 3 and 4 values when the values of its parameter change,it remains constant and unchanged concerning its order or number of *** methodology of determining the metric basis or locating set is based on the distances of a ***,we proved the main theorems in distance forms.

关键词： Klein bottle metric basis resolving set metric dimension

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ON CLASSES OF REGULAR GRAPHS WITH CONSTANT METRIC DIMENSION

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Acta Mathematica Scientia 2013年第1期33卷 187-206页

作者： Muhammad IMRAN Syed Ahtsham ul Haq BOKHARY Ali AHMAD Andrea SEMANIOV-FENOVíKOV Centre for Advanced Mathematics and Physics(CAMP) National University of Sciences and Technology(NUST) Center for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya University College of Computer and Information System Jazan University Department of Applied Mathematics and Informatics Technical University

In this paper, we are dealing with the study of the metric dimension of some classes of regular graphs by considering a class of bridgeless cubic graphs called the flower snarks Jn, a class of cubic convex polytopes considering the open problem raised in [M. Imran et al., families of plane graphs with constant metric dimension, Utilitas Math., in press] and finally Harary graphs H5,n by partially answering to an open problem proposed in Ⅱ. Javaid et al., Families of regular graphs with constant metric dimension, Utilitas Math., 2012, 88： 43-57]. We prove that these classes of regular graphs have constant metric dimension.

关键词： metric dimension basis resolving set cubic graph flower snark convexpolytope

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On the Metric Dimension of Barycentric Subdivision of Cayley Graphs

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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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A Comparison between the Metric Dimension and Zero Forcing Number of Trees and Unicyclic Graphs

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Acta Mathematica Sinica,English Series 2017年第6期33卷 731-747页

作者： Linda EROH Cong X.KANG Eunjeong YI University of Wisconsin Oshkosh Oshkosh WI 54901 USA Texas A & M University at Galveston Galveston TX 77553 USA

The metric dimension dim（G） of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z（G） of a graph G is the minimum eardinality of a set S of black vertices （whereas vertices in V（G）／S are colored white） such that V（G） is turned black after finitely many applications of ＂the color-change rule＂： a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim（T） ≤Z（T） for a tree T, and that dim（G）≤Z（G）＋I if G is a unicyclic graph; along the way, we characterize trees T attaining dim（T） = Z（T）. For a general graph G, we introduce the ＂cycle rank conjecture＂. We conclude with a proof of dim（T） - 2 ≤ dim（T ＋ e） ≤dim（T）＋ 1 for e∈ E（T）.

关键词： Distance resolving set metric dimension zero forcing set zero forcing number tree,unicyclic graph cycle rank

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Fault-Tolerant Resolvability of Certain Crystal Structures

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Applied Mathematics 2016年第7期7卷 599-604页

作者： Sathish Krishnan Bharati Rajan Research and Development Centre Bharathiar University Coimbatore India Department of Mathematics DMI College of Engineering Chennai India Department of Mathematics Loyola College Chennai India School of Mathematical and Physical Sciences Faculty of Science and IT The University of Newcastle Callaghan Australia

An ordered set W of vertices of a graph G is called a resolving set, if all the vertices of G are uniquely determined by the vector of distances to the vertices in W. The metric dimension of G is the minimum cardinality of a resolving set of G. A resolving set W for G is fault-tolerant if W\{v} is also a resolving set, for each v in W, and the fault-tolerant metric dimension of G is the minimum cardinality of such a set. In this paper we determine the metric dimension and fault-tolerant metric dimension problems for the graphs of certain crystal structures.

关键词： resolving set Metric Dimension Fault-Tolerant metric Dimension Crystal Structures Bismuth Tri-Iodide Lead Chloride Quartz

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