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.

The Kemeny’s Constant and Spanning Trees of Hexagonal Ring Network

Spanning tree(τ)has an enormous application in computer science and chemistry to determine the geometric and dynamics analysis of compact polymers.In the field of medicines,it is helpful to recognize the epidemiology of hepatitis C virus(HCV)infection.On the other hand,Kemeny’s constant(Ω)is a beneficial quantifier characterizing the universal average activities of a Markov chain.This network invariant infers the expressions of the expected number of time-steps required to trace a randomly selected terminus state since a fixed beginning state si.Levene and Loizou determined that the Kemeny’s constant can also be obtained through eigenvalues.Motivated by Levene and Loizou,we deduced the Kemeny’s constant and the number of spanning trees of hexagonal ring network by their normalized Laplacian eigenvalues and the coefficients of the characteristic polynomial.Based on the achieved results,entirely results are obtained for the M鯾ius hexagonal ring network.

Comparative Study of Valency-Based Topological Descriptor for Hexagon Star Network

A class of graph invariants referred to today as topological indices are inefficient progressively acknowledged by scientific experts and others to be integral assets in the depiction of structural phenomena.The structure of an interconnection network can be represented by a graph.In the network,vertices represent the processor nodes and edges represent the links between the processor nodes.Graph invariants play a vital feature in graph theory and distinguish the structural properties of graphs and networks.A topological descriptor is a numerical total related to a structure that portray the topology of structure and is invariant under structure automorphism.There are various uses of graph theory in the field of basic science.The main notable utilization of a topological descriptor in science was by Wiener in the investigation of paraffin breaking points.In this paper we study the topological descriptor of a newly design hexagon star network.More preciously,we have computed variation of the Randic0 R0,fourth Zagreb M4,fifth Zagreb M5,geometric-arithmetic GA;atom-bond connectivity ABC;harmonic H;symmetric division degree SDD;first redefined Zagreb,second redefined Zagreb,third redefined Zagreb,augmented Zagreb AZI,Albertson A;Irregularity measures,Reformulated Zagreb,and forgotten topological descriptors for hexagon star network.In the analysis of the quantitative structure property relationships(QSPRs)and the quantitative structure activity relationships(QSARs),graph invariants are important tools to approximate and predicate the properties of the biological and chemical compounds.We also gave the numerical and graphical representations comparisons of our different results.

Optimized Evaluation of Mobile Base Station by Modern Topological Invariants

Due to a tremendous increase in mobile traffic,mobile operators have started to restructure their networks to offload their traffic.Newresearch directions will lead to fundamental changes in the design of future Fifthgeneration(5G)cellular networks.For the formal reason,the study solves the physical network of the mobile base station for the prediction of the best characteristics to develop an enhanced network with the help of graph theory.Any number that can be uniquely calculated by a graph is known as a graph invariant.During the last two decades,innumerable numerical graph invariants have been portrayed and used for correlation analysis.In any case,no efficient assessment has been embraced to choose,how much these invariants are connected with a network graph.This paper will talk about two unique variations of the hexagonal graph with great capability of forecasting in the field of optimized mobile base station topology in setting with physical networks.Since K-banhatti sombor invariants(KBSO)and Contrharmonic-quadratic invariants(CQIs)are newly introduced and have various expectation characteristics for various variations of hexagonal graphs or networks.As the hexagonal networks are used in mobile base stations in layered,forms called honeycomb.The review settled the topology of a hexagon of two distinct sorts with two invariants KBSO and CQIs and their reduced forms.The deduced outcomes can be utilized for the modeling of mobile cellular networks,multiprocessors interconnections,microchips,chemical compound synthesis and memory interconnection networks.The results find sharp upper bounds and lower bounds of the honeycomb network to utilize the Mobile base station network(MBSN)for the high load of traffic and minimal traffic also.

The criterion of anomalous slip at OK in body centered cubic metals

It is generally accepted that anomalous slip(AS) takes place by hexagonal dislocation networks(HDNs) in body centered cubic(BCC) metals,but the role of the HDN formation process in AS has rarely been investigated so far.In this work,the critical yield conditions of the HDNs and isolated dislocations were first calculated,respectively,by molecular statics simulations in two BCC metals.Based on these data,a novel mechanism,entitled as the "conjugated dislocation sources"(CDS),to analyze the formation of the HDNs was proposed for the first time and then incorporated into the criterion of the occurrence of AS.Our prediction is in agreement with experimental observations.Contrary to previous study,it has been revealed that the multiplication of isolated screw dislocations involved in AS has to be considered for correctly understanding the AS origin.