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.

ON THE GRAPHS OF PRODUCTS OF CONTINUOUS FUNCTIONS AND FRACTAL DIMENSIONS

In this paper,we consider the graph of the product of continuous functions in terms of Hausdorff and packing dimensions.More precisely,we show that,given a real number 1≤β≤2,any real-valued continuous function in C([0,1])can be decomposed into a product of two real-valued continuous functions,each having a graph of Hausdorff dimensionβ.In addition,a product decomposition result for the packing dimension is obtained.This work answers affirmatively two questions raised by Verma and Priyadarshi[14].

The Dimensions of Graphs and Constructions of Their Bases

Let G(V,E) be a connected graph and W{w 1,w 2,…,w k} an ordered set of V. Given v∈V, the representation of v with respect to W is the k-vector r(v|W)(d(v,w 1),d(v,w 2),…,d(v,w k)). The set W is a resolving set of G if r(u|W)r(v|W) implies that uv for all pairs {u,v} of vertices of G. The resolving set of G with the smallest cardinality is called a basis of G. The dimension of G, dim (G), is the cardinality of a basis for G. The bound of a Cartesian product of a connected graph H and a path P k was reached: dim(H)≤dim(H×P k)≤dim(H)+1. Then, the dimension value of some graphs was given. At last, the constructions of some graphs’ bases were showed.

THE HAUSDORFF DIMENSION AND MEASURE IN GENERALIZED GEOMETRIC GRAPH DIRECTED CONSTRUCTION

Abthors introduce the notation of generalized geometric constructions in Rm generated by a directed graph G and by a sequence of similarity ratios which are labelled with the edges of this graph. In this paper, it is obtained the Hausdorff dimension and measure of this construction object for some cases.

CONNECTION BETWEEN THE ORDER OF FRACTIONAL CALCULUS AND FRACTIONAL DIMENSIONS OF A TYPE OF FRACTAL FUNCTIONS

The linear relationship between fractal dimensions of a type of generalized Weierstrass functions and the order of their fractional calculus has been proved. The graphs and numerical results given here further indicate the corresponding relationship.

Some results on entropy dimension for non-autonomous systems

In this paper,the preimage branch t-entropy and entropy dimension for nonautonomous systems are studied and some systems with preimage branch t-entropy zero are introduced.Moreover,formulas calculating the s-topological entropy of a sequence of equi-continuous monotone maps on the unit circle are given.Finally,examples to show that the entropy dimension of non-autonomous systems can be attained by any positive number s are constructed.

DECIMAL SET WITH ZERO MEASURE AND FULL HAUSDORFF DIMENSION

LetFλ ={x∈ (0,1){2nx} ≥1/2k,n∈ z+}, z++ = {0,1,2,3,...}, k∈ N;F = U∞k=1Fλ be a decimal set in (0, 1), where {2nx} is the fractional part of a number 2nx. In this note, we prove that dirnнF = 1 and Н1(F) = 0, where dimн is Hausdr off dimension, and Н1(F) is the Hausdorff measure of F.

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.

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.

Hausdorff Dimension of Range and Graph for General Markov Processes

We establish the Hausdorff dimension of the graph of general Markov processes on Rd based on some probability estimates of the processes staying or leaving small balls in small time.In particular,our results indicate that,for symmetric diffusion processes(withα=2)or symmetricα-stable-like processes(withα∈(0,2))on Rd,it holds almost surely that dimH GrX([0,1])=1{α<1}+(2−1/α)1{α≥1,d=1}+(d∧α)1{α≥1,d≥2}.We also systematically prove the corresponding results about the Hausdorff dimension of the range of the processes.

THE REGULARITY OF RANDOM GRAPH DIRECTED SELF-SIMILAR SETS

A set in Rd is called regular if its Hausdorff dimension coincides with its upper box counting dimension. It is proved that a random graph-directed self-similar set is regular a.e..

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.

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）.

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 Strong Metric Dimension of Graphs and Their Complements

A vertex x in a graph G strongly resolves a pair of vertices v, w if there exists a shortest x-w path containing v or a shortest x-v path containing w in G. A set of vertices SV（G） is a strong resolving set of G if every pair of distinct vertices of G is strongly resolved by some vertex in S. The strong metric dimension of G, denoted by sdim（G）, is the minimum cardinality over all strong resolving sets of G. For a connected graph G of order n≥2, we characterize G such that sdim（G） equals 1, n-1, or n-2, respectively. We give a Nordhaus–Gaddum-type result for the strong metric dimension of a graph and its complement： for a graph G and its complement G, each of order n≥4 and connected, we show that 2≤sdim（G）＋sdim（G）≤2（ n-2）. It is readily seen that sdim（G）＋sdim（G）=2 if and only if n=4; we show that, when G is a tree or a unicyclic graph, sdim（G）＋sdim（G）=2（n 2） if and only if n=5 and G ～=G ～=C5, the cycle on five vertices. For connected graphs G and G of order n≥5, we show that 3≤sdim（G）＋sdim（G）≤2（n-3） if G is a tree; we also show that 4≤sdim（G）＋sdim（G）≤2（n-3） if G is a unicyclic graph of order n≥6. Furthermore, we characterize graphs G satisfying sdim（G）＋sdim（G）=2（n-3） when G is a tree or a unicyclic graph.

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

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）.

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.

历史学维度下中国绘画的三种语图关系

在中国绘画史发展过程中,绘画和语言从来都是互为表里、相互补充的关系。从历史学维度涉及的语言和图像之间的关系角度,探讨中国绘画的三种语图关系。具体而言,中国绘画的语图关系主要包括语图一体、语图分体和语图合体这三大部分,包括分体关系、合体关系、直接影响古今文学艺术发展的特点及形成,并对当今艺术创作及发展有启示价值。

基于MFLME的转子故障特征降维方法

针对故障特征维数过高导致故障的分类与辨识性能不佳的现状,提出一种基于中值特征线多图嵌入(median feature line multi-graph embedding,简称MFLME)的故障数据集降维算法。首先,将样本点到特征空间的投影度量改进为中值度量,削弱算法的外推误差;其次,通过定义近邻特征线图和远邻特征线图,减少异类样本的混淆,扩大类别间距,为后续故障的分类决策降低难度;最后,利用两个不同的转子故障模拟实验对算法性能进行验证。结果表明,该算法能降低故障分类难度,提升故障辨识准确率。