Bounds on Fractional-Based Metric Dimension of Petersen Networks

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 network.In 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≥7.For n≥9.The limiting values of LFMD for GP(n,3)are also obtained as 1(bounded)if n approaches to infinity.

Metric Basis of Four-Dimensional Klein Bottle

The Metric of a graph plays an essential role in the arrangement of different dimensional structures and finding their basis in various terms.The 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 vertices.This set of selected vertices is known as the metric basis of a graph.In 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 network.Due to the vast applications of this concept in computer science,optimization problems,and also in chemistry enormous research has been conducted.To 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 parameters.Although 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 vertices.The methodology of determining the metric basis or locating set is based on the distances of a graph.Therefore,we proved the main theorems in distance forms.

Binary Archimedes Optimization Algorithm for Computing Dominant Metric Dimension Problem

In this paper,we consider the NP-hard problem of finding the minimum dominant resolving set of graphs.A vertex set B of a connected graph G resolves G if every vertex of G is uniquely identified by its vector of distances to the vertices in B.A resolving set is dominating if every vertex of G that does not belong to B is a neighbor to some vertices in B.The dominant metric dimension of G is the cardinality number of the minimum dominant resolving set.The dominant metric dimension is computed by a binary version of the Archimedes optimization algorithm(BAOA).The objects of BAOA are binary encoded and used to represent which one of the vertices of the graph belongs to the dominant resolving set.The feasibility is enforced by repairing objects such that an additional vertex generated from vertices of G is added to B and this repairing process is iterated until B becomes the dominant resolving set.This is the first attempt to determine the dominant metric dimension problem heuristically.The proposed BAOA is compared to binary whale optimization(BWOA)and binary particle optimization(BPSO)algorithms.Computational results confirm the superiority of the BAOA for computing the dominant metric dimension.

On Cycle Related Graphs with Constant Metric Dimension

If G is a connected graph, the distance d (u,v) between two vertices u,v ∈ V(G) is the length of a shortest path between them. Let W = {w1, w2, ..., wk} be an ordered set of vertices of G and let v be a vertex of G . The repre-sentation r(v|W) of v with respect to W is the k-tuple (d(v,w1), d(v,w2), …, d(v,wk)). . If distinct vertices of G have distinct representations with respect to W , then W is called a resolving set or locating set for G. A re-solving set of minimum cardinality is called a basis for G and this cardinality is the metric dimension of G , denoted by dim (G). A family ? of connected graphs is a family with constant metric dimension if dim (G) is finite and does not depend upon the choice of G in ?. In this paper, we show that dragon graph denoted by Tn,m and the graph obtained from prism denoted by 2Ck + {xkyk} have constant metric dimension.

ON CLASSES OF REGULAR GRAPHS WITH CONSTANT METRIC DIMENSION

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.

Almost Injective Mappings of Totally Bounded Metric Spaces into Finite Dimensional Euclidean Spaces

Let χ= be a metric space and let ε be a positive real number. Then a function f: X→Y is defined to be an ε-map if and only if for all y∈Y, the diameter of f-1(y)?is at most ε. In Theorem 10 we will give a new proof for the following well known fact: if χ is totally bounded, then for all ε there exists a finite number n and a continuous ε-map fε: X→Rn (here Rn is the usual n-dimensional Euclidean space endowed with the Euclidean metric). If ε is “small”, then fε is “almost injective”;and still exists even if χ has infinite covering dimension (in this case, n depends on ε, of course). Contrary to the known proofs, our proof technique is effective in the sense, that it allows establishing estimations for n in terms of ε and structural properties of χ.

Edge Metric Dimension of Honeycomb and Hexagonal Networks for IoT

Wireless Sensor Network(WSN)is considered to be one of the fundamental technologies employed in the Internet of things(IoT);hence,enabling diverse applications for carrying out real-time observations.Robot navigation in such networks was the main motivation for the introduction of the concept of landmarks.A robot can identify its own location by sending signals to obtain the distances between itself and the landmarks.Considering networks to be a type of graph,this concept was redefined as metric dimension of a graph which is the minimum number of nodes needed to identify all the nodes of the graph.This idea was extended to the concept of edge metric dimension of a graph G,which is the minimum number of nodes needed in a graph to uniquely identify each edge of the network.Regular plane networks can be easily constructed by repeating regular polygons.This design is of extreme importance as it yields high overall performance;hence,it can be used in various networking and IoT domains.The honeycomb and the hexagonal networks are two such popular mesh-derived parallel networks.In this paper,it is proved that the minimum landmarks required for the honeycomb network HC(n),and the hexagonal network HX(n)are 3 and 6 respectively.The bounds for the landmarks required for the hex-derived network HDN1(n)are also proposed.

Fault-Tolerant Resolvability of Certain Crystal Structures

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.

Metric and Upper Dimension of Extended Annihilating-Ideal Graphs

The metric dimension problem is called navigation problem due to its application to robot navigation in space.Further this concept has wide applications in motion planning,sonar and loran station,and so on.In 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)),respectively.Here 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.

THE DIMENSION FOR RANDOM SUB-SELF-SIMILAR SET

In this article, the Hausdorff dimension and exact Hausdorff measure function of any random sub-self-similar set are obtained under some reasonable conditions. Several examples are given at the end.

Metric Identification of Vertices in Polygonal Cacti

The distance between two vertices u and v in a connected graph G is the number of edges lying in a shortest path(geodesic)between them.A vertex x of G performs the metric identification for a pair(u,v)of vertices in G if and only if the equality between the distances of u and v with x implies that u=v(That is,the distance between u and x is different from the distance between v and x).The minimum number of vertices performing the metric identification for every pair of vertices in G defines themetric dimension of G.In this paper,we performthemetric identification of vertices in two types of polygonal cacti:chain polygonal cactus and star polygonal cactus.

Metric-Based Resolvability of Quartz Structure

Silica has three major varieties of crystalline. Quartz is the main andabundant ingredient in the crust of our earth. While other varieties are formedby the heating of quartz. Silica quartz is a rich chemical structure containingenormous properties. Any chemical network or structure can be transformedinto a graph, where atoms become vertices and the bonds are converted toedges, between vertices. This makes a complex network easy to visualize towork on it. There are many concepts to work on chemical structures in termsof graph theory but the resolvability parameters of a graph are quite advanceand applicable topic. Resolvability parameters of a graph is a way to getting agraph into unique form, like each vertex or edge has a unique identification bymeans of some selected vertices, which depends on the distance of vertices andits pattern in a particular graph. We have dealt some resolvability parametersof SiO2 quartz. We computed the resolving set for quartz structure and itsvariants, wherein we proved that all the variants of resolvability parameters ofquartz structures are constant and do not depend on the order of the graph.

基于全局-局部欧拉弹性判别投影的旋转机械故障诊断方法

故障诊断方法通常对异常值敏感,并且难以同时提取全局和局部判别信息,从而导致低维判别特征子集类间分离性不佳,针对该问题提出了一种基于全局-局部欧拉弹性判别投影(global-local euler elastic discriminant projection,GLEEDP)的旋转机械故障诊断方法。该方法通过余弦度量将高维故障特征映射到欧拉表示空间,扩大异类故障样本间的差异,然后在该空间中构建了基于全局、局部及类间散布三个目标函数的最优化模型,实现了在保持整体结构的基础上,进一步提高低维判别特征子集的局部类内聚集性和全局类间分离性。在轴承和齿轮箱两个机械故障数据集上的试验结果表明,所提方法可以有效发掘故障判别信息,具有优越的故障诊断性能。

The Simultaneous Fractional Dimension of Graph Families

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 G.For 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∈C.The 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 family.We obtain max1≤i≤k{dimf(Gi)}≤Sd_(f)(C)≤min{∑k i=1 dimf(Gi),|V|/2},where both bounds are sharp.We 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 graphs.We also obtain some results on the simultaneous fractional dimension of a graph and its complement.

Minimal Doubly Resolving Sets of Certain Families of Toeplitz Graph

The doubly resolving sets are a natural tool to identify where diffusion occurs in a complicated network.Many 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 networks.It 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 state.As a result,the source localization problem is to find the number of nodes in the network that best explains the observed diffusion.This 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 detection.This paper aims to investigate the minimal doubly resolving set for certain families of Toeplitz graph Tn(1,t),for t≥2 and n≥t+2.We 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 dimension.Also,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.

关于非交换群的非交换图的度量维数

给定一个非交换群,该群的非交换图以该群所有非中心元素构成的集合为顶点集,两个不同的顶点x和y相邻的充分必要条件是xy≠yx.文章研究非交换群的非交换图的度量维数问题,确定了二面体群的非交换图的度量维数,且对任意非交换群的非交换图的度量维数给出了紧的上下界.

平面图γ_(n)的混合度量维数

图的度量维数是图论与组合优化中的一个重要问题.混合度量维数是度量维数的一个变形.给出了平面图γ_(n)的一个混合度量生成集,得到了平面图γ_(n)的混合度量维数的上界.

On the Constant Metric Dimension of Generalized Petersen Graphs P(n,4)

In this paper, we consider the family of generalized Petersen graphs P（n,4）. We prove that the metric dimension of P（n, 4） is 3 when n = 0 （mod 4）, and is 4 when n = 4k ＋ 3 （k is even）.For n = 1,2 （mod 4） and n = 4k ＋ 3 （k is odd）, we prove that the metric dimension of P（n,4） is bounded above by 4. This shows that each graph of the family of generalized Petersen graphs P（n, 4） has constant metric dimension.

On the Metric Dimension of Barycentric Subdivision of Cayley Graphs

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）.

The Fractional Metric Dimension of Permutation Graphs

Let G =（V（G）, E（G）） be a graph with vertex set V（G） and edge set E（G）. For two distinct vertices x and y of a graph G, let RG{x, y} denote the set of vertices z such that the distance from x to z is not equa l to the distance from y to z in G. For a function g defined on V（G） and for U V（G）, let g（U） =∑s∈Ug（s）. A real-valued function g ： V（G） → [0, 1] is a resolving function of G if g（RG{x, y}） ≥ 1 for any two distinct vertices x, y ∈ V（G）. The fractional metric dimension dimf（G）of a graph G is min{g（V（G）） ： g is a resolving function of G}. Let G1 and G2 be disjoint copies of a graph G, and let σ ： V（G1） → V（G2） be a bijection. Then, a permutation graph Gσ =（V, E） has the vertex set V = V（G1） ∪ V（G2） and the edge set E = E（G1） ∪ E（G2） ∪ {uv ｜ v = σ（u）}. First,we determine dimf（T） for any tree T. We show that 1 〈 dimf（Gσ） ≤1/2（｜V（G）｜ ＋ ｜S（G）｜） for any connected graph G of order at least 3, where S（G） denotes the set of support vertices of G. We also show that, for any ε 〉 0, there exists a permutation graph Gσ such that dimf（Gσ）- 1 〈 ε. We give examples showing that neither is there a function h1 such that dimf（G） 〈 h1（dimf（Gσ）） for all pairs（G, σ）, nor is there a function h2 such that h2（dimf（G）） 〉 dimf（Gσ） for all pairs（G, σ）. Furthermore,we investigate dimf（Gσ） when G is a complete k-partite graph or a cycle.