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Bounds on Fractional-Based metric dimension of Petersen Networks

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Computer Modeling in Engineering & Sciences 2023年第6期135卷 2697-2713页

作者： Dalal Awadh Alrowaili Mohsin Raza Muhammad Javaid Mathematics Department College of ScienceJouf UniversityP.O.Box:2014SakakaSaudi Arabia Department of Mathematics School of ScienceUniversity of Management and TechnologyLahorePakistan

The problem of investigating the minimum set of landmarks consisting of auto-machines(Robots)in a connected network is studied with the concept of location number ormetric dimension of this *** this paper,we study the latest type of metric dimension called as local fractional metric dimension(LFMD)and find its upper bounds for generalized Petersen networks GP(n,3),where n≥*** n≥*** limiting values of LFMD for GP(n,3)are also obtained as 1(bounded)if n approaches to infinity.

关键词： metric dimension local fractional metric dimension Petersen network local resolving neighborhoods

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

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

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

metric and Upper dimension of Extended Annihilating-Ideal Graphs

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Algebra Colloquium 2024年第2期31卷 221-238页

作者： S.Nithya G.Elavarasi Genghua Fan Department of Mathematics St.Xavier's College(Autonomous)Palayamkottai 627002Tamil NaduIndia Afiliated to Manonmaniam Sundaranar University Abishekapatti Tirunelveli 627012Tamil NaduIndia PG and Research Department of Mathematics St.Xavier's College(Autonomous)Palayamkottai 627002Tamil NaduIndia Affiliated to Manonmaniam Sundaranar University Abishekapatti Tirunelveli 627012Tamil NaduIndia 不详

The metric dimension problem is called navigation problem due to its application to robot navigation in *** this concept has wide applications in motion planning,sonar and loran station,and so *** this paper,we study certain results on the metric dimension,upper dimension and resolving number of extended annihilating-ideal graph EAG(R)associated to a commutative ring R,denoted by dim M(EAG(R)),dim+(EAG(R))and res(EAG(R)),*** we prove the finiteness conditions of dim M(EAG(R))and dim+(EAG(R)).In addition,we characterize dim M(EAG(R)),dim+(EAG(R))and res(EAG(R))for artinian rings and the direct product of rings.

关键词： extended annihilating-ideal graph metric dimension upper dimension resolving number

The Simultaneous Fractional dimension of Graph Families

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Acta Mathematica Sinica,English Series 2023年第8期39卷 1425-1441页

作者： Cong X.KANG Iztok PETERIN Eunjeong YI Texas A&M University at Galveston GalvestonTX 77553USA University of Maribor FEECS Smetanova 172000 MariborSlovenia

For a connected graph G with vertex set V,let RG{x,y}={z∈V:dG(x,z)≠dG(y,z)}for any distinct x,y∈V,where dG(u,w)denotes the length of a shortest uw-path in *** a real-valued function g defined on V,let g(V)=∑s∈V g(s).Let C={G_(1),G_(2),...,G_(k)}be a family of connected graphs having a common vertex set V,where k≥2 and|V|≥3.A real-valued function h:V→[0,1]is a simultaneous resolving function of C if h(RG{x,y})≥1 for any distinct vertices x,y∈V and for every graph G∈*** simultaneous fractional dimension,Sdf(C),of C is min{h(V):h is a simultaneous resolving function of C}.In this paper,we initiate the study of the simultaneous fractional dimension of a graph *** obtain max1≤i≤k{dimf(Gi)}≤Sd_(f)(C)≤min{∑k i=1 dimf(Gi),|V|/2},where both bounds are *** characterize C satisfying Sdf(C)=1,examine C satisfying Sdf(C)=|V|/2,and determine Sdf(C)when C is a family of vertex-transitive *** also obtain some results on the simultaneous fractional dimension of a graph and its complement.

关键词： metric dimension fractional metric dimension resolving function simultaneous(metric)dimension simultaneous fractional(metric)dimension

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The Commuting Graphs on Dicyclic Groups

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Algebra Colloquium 2020年第4期27卷 799-806页

作者： Jing Chen Lang Tang School of Mathematics and Computational Science Hunan First Normal University Changsha 410073China

For a group G and a non-empty subsetΩof G,the commuting graph C(G,Ω)ofΩis a graph whose vertex set isΩand any two vertices are adjacent if and only if they commute in *** T4n=(a,b|a^(2)n=b^(4)=1,an=b2,b^(−1)ab=a^(−1)),the dicyclic group of order 4n(n≥3),which is also known as the generalized quaternion *** mainly investigate the properties and metric dimension of the commuting graphs on the dicyclic group T4n.

关键词： commuting graph dicyclic group perfect matching metric dimension

Resolvability in Circulant Graphs

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Acta Mathematica Sinica,English Series 2012年第9期28卷 1851-1864页

作者： Muhammad SALMAN Imran JAVAID Muhammad Anwar CHAUDHRY Center for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya University

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v E V（G） there is a vertex w E W such that d（u, w） ≠ d（v, w）. A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim（G）. For a vertex u of G and a subset S of V（G）, the distance between u and S is the number minses d（u,s）. A k-partition II = {$1,$2,..., Sk} of V（G） is called a resolving partition if for every two distinct vertices u, v E V（G） there is a set Si in H such that d（u, Si） ≠ d（v, Si）. The minimum k for which there is a resolving k-partition of V（G） is called the partition dimension of G, denoted by pd（G）. The circulant graph is a graph with vertex set Zn, an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i - j （rood n） E C, where C C Zn has the property that C = -C and 0 ￠ C. The circulant graph is denoted by Xn,△ where A = ｜C｜. In this paper, we study the metric dimension of a family of circulant graphs Xn,3 with connection set C = {1, 3, n - 1} and prove that dim（Xn,3） is independent of choice of n by showing that dim（Xn,a） = {3 for all n ≡0 （mod 4）,4 for all n≡2 （mod4）.We also study the partition dimension of a family of circulant graphs Xm,4 with connection set C = {±1, ±2} and prove that pd（Xn,4） is independent of choice of n and show that pd（X5,4） = 5 and pd（Xn,a） ={ 3 for all odd n≥9,4 for all even n≥6 and n=7.

关键词： Circulant graphs metric dimension partition dimension

Minimal Doubly Resolving Sets of Certain Families of Toeplitz Graph

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Computer Modeling in Engineering & Sciences 2023年第6期135卷 2681-2696页

作者： Muhammad Ahmad Fahd Jarad Zohaib Zahid Imran Siddique Department of Mathematics University of Management and TechnologyLahore54770Pakistan Department of Mathematics Cankaya UniversityEtimesgutAnkara06790Turkey Department of Medical Research China Medical University HospitalChina Medical UniversityTaichung40402Taiwan

The doubly resolving sets are a natural tool to identify where diffusion occurs in a complicated *** realworld phenomena,such as rumour spreading on social networks,the spread of infectious diseases,and the spread of the virus on the internet,may be modelled using information diffusion in *** is obviously impractical to monitor every node due to cost and overhead limits because there are too many nodes in the network,some of which may be unable or unwilling to send information about their *** a result,the source localization problem is to find the number of nodes in the network that best explains the observed *** problem can be successfully solved by using its relationship with the well-studied related minimal doubly resolving set problem,which minimizes the number of observers required for accurate *** paper aims to investigate the minimal doubly resolving set for certain families of Toeplitz graph Tn(1,t),for t≥2 and n≥t+*** come to the conclusion that for Tn(1,2),the metric and double metric dimensions are equal and for Tn(1,4),the double metric dimension is exactly one more than the metric ***,the double metric dimension for Tn(1,3)is equal to the metric dimension for n=5,6,7 and one greater than the metric dimension for n≥8.

关键词： Family of Toeplitz graph resolving sets metric dimension doubly resolving sets double metric dimension