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.

Computing Connected Resolvability of Graphs Using Binary Enhanced Harris Hawks Optimization

In this paper,we consider the NP-hard problem offinding the minimum connected 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 ver-tices in B.A resolving set B of G is connected if the subgraph B induced by B is a nontrivial connected subgraph of G.The cardinality of the minimal resolving set is the metric dimension of G and the cardinality of minimum connected resolving set is the connected metric dimension of G.The problem is solved heuristically by a binary version of an enhanced Harris Hawk Optimization(BEHHO)algorithm.This is thefirst attempt to determine the connected resolving set heuristically.BEHHO combines classical HHO with opposition-based learning,chaotic local search and is equipped with an S-shaped transfer function to convert the contin-uous variable into a binary one.The hawks of BEHHO are binary encoded and are used to represent which one of the vertices of a graph belongs to the connected resolving set.The feasibility is enforced by repairing hawks such that an addi-tional node selected from V\B is added to B up to obtain the connected resolving set.The proposed BEHHO algorithm is compared to binary Harris Hawk Optimi-zation(BHHO),binary opposition-based learning Harris Hawk Optimization(BOHHO),binary chaotic local search Harris Hawk Optimization(BCHHO)algorithms.Computational results confirm the superiority of the BEHHO for determining connected metric dimension.

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.

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.

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.

FUNCTIONAL ANALYSIS METHOD FOR THE M/G/1 QUEUEING MODEL WITH OPTIONAL SECOND SERVICE

By studying the spectral properties of the underlying operator corresponding to the M/G/1 queueing model with optional second service we obtain that the time-dependent solution of the model strongly converges to its steady-state solution. We also show that the time-dependent queueing size at the departure point converges to the corresponding steady-state queueing size at the departure point.

Tripling Construction for Large Sets of Resolvable Directed Triple Systems

In this paper, we first define a doubly transitive resolvable idempotent quasigroup （DTRIQ）, and show that aDTRIQ of order v exists if and only ifv ≡0（mod3） and v ≠ 2（mod4）. Then we use DTRIQ to present a tripling construction for large sets of resolvable directed triple systems, which improves an earlier version of tripling construction by Kang （J. Combin. Designs, 4 （1996）, 301-321）. As an application, we obtain an LRDTS（4·3^n） for any integer n ≥ 1, which provides an infinite family of even orders.

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）.

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）.

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.