Lower and Upper Bounds of Fractional Metric Dimension of Connected Networks

The distance centric parameter in the theory of networks called by metric dimension plays a vital role in encountering the distance-related problems for the monitoring of the large-scale networks in the various fields of chemistry and computer science such as navigation, image processing, pattern recognition, integer programming, optimal transportation models and drugs discovery. In particular, it is used to find the locations of robots with respect to shortest distance among the destinations, minimum consumption of time, lesser number of the utilized nodes, and to characterize the chemical compounds, having unique presentations in molecular networks. After the arrival of its weighted version, known as fractional metric dimension, the rectification of distance-related problems in the aforementioned fields has revived to a great extent. In this article, we compute fractional as well as local fractional metric dimensions of web-related networks called by subdivided QCL, 2-faced web, 3-faced web, and antiprism web networks. Moreover, we analyse their final results using 2D and 3D plots.

Boundedness of Convex Polytopes Networks via Local Fractional Metric Dimension

Metric dimension is one of the distance-based parameter which is frequently used to study the structural and chemical properties of the different networks in the various fields of computer science and chemistry such as image processing, pattern recognition, navigation, integer programming, optimal transportation models, and drugs discovery. In particular, it is used to find the locations of robots with respect to shortest distance among the destinations, minimum consumption of time, and lesser number of the utilized nodes and to characterize the chemical compounds having unique presentation in molecular networks. The fractional metric dimension being a latest developed weighted version of the metric dimension is used in the distance-related problems of the aforementioned fields to find their nonintegral optimal solutions. In this paper, we have formulated the local resolving neighborhoods with their cardinalities for all the edges of the convex polytopes networks to compute their local fractional metric dimensions in the form of exact values and sharp bounds. Moreover, the boundedness of all the obtained results is also proved.

Computing LF-Metric Dimension of Generalized Gear Networks

The parameter of distance in the theory of networks plays a key role to study the different structural properties of the understudy networks or graphs such as symmetry, assortative, connectivity, and clustering. For the purpose, with the help of the parameter of distance, various types of metric dimensions have been defined to find the locations of machines (or robots) with respect to the minimum consumption of time, the shortest distance among the destinations, and the lesser number of utilized nodes as places of the objects. In this article, the latest derived form of metric dimension called as LF-metric dimension is studied, and various results for the generalized gear networks are obtained in the form of exact values and sharp bounds under certain conditions. The LF-metric dimension of some particular cases of generalized gear networks (called as generalized wheel networks) is also illustrated. Moreover, the bounded and unboundedness of the LF-metric dimension for all obtained results is also presented.

The Vertex-Edge Resolvability of Some Wheel-Related Graphs

A vertex w ∈ V H distinguishes (or resolves) two elements (edges or vertices) a , z ∈ V H ∪ E H if d w , a ≠ d w , z . A set W m of vertices in a nontrivial connected graph H is said to be a mixed resolving set for H if every two different elements (edges and vertices) of H are distinguished by at least one vertex of W m . The mixed resolving set with minimum cardinality in H is called the mixed metric dimension (vertex-edge resolvability) of H and denoted by m dim H . The aim of this research is to determine the mixed metric dimension of some wheel graph subdivisions. We specifically analyze and compare the mixed metric, edge metric, and metric dimensions of the graphs obtained after the wheel graphs’ spoke, cycle, and barycentric subdivisions. We also prove that the mixed resolving sets for some of these graphs are independent.

On the Upper Bounds of Fractional Metric Dimension of Symmetric Networks

Distance-based numeric parameters play a pivotal role in studying the structural aspects of networks which include connectivity, accessibility, centrality, clustering modularity, complexity, vulnerability, and robustness. Several tools like these also help to resolve the issues faced by the different branches of computer science and chemistry, namely, navigation, image processing, biometry, drug discovery, and similarities in chemical compounds. For this purpose, in this article, we are considering a family of networks that exhibits rotationally symmetric behaviour known as circular ladders consisting of triangular, quadrangular, and pentagonal faced ladders. We evaluate their upper bounds of fractional metric dimensions of the aforementioned networks.

Metric Dimensions Of Graphs

In this article, some kinds of triple belongs to metric dimensions are defined. Some classes of graphs in the matter of these kinds, are studied and the relation amid these kinds are considered. The kind of having equivalency amid these notions and some classes of graphs, is obtained. The kind of locating some vertices by some vertices when the number of locating vertices is increased, has the key role to analyze the classes of graphs, general graphs, and graph's parameters.

Bounds of Fractional Metric Dimension and Applications with Grid-Related Networks

Metric dimension of networks is a distance based parameter that is used to rectify the distance related problems in robotics, navigation and chemical strata. The fractional metric dimension is the latest developed weighted version of metric dimension and a generalization of the concept of local fractional metric dimension. Computing the fractional metric dimension for all the connected networks is an NP-hard problem. In this note, we find the sharp bounds of the fractional metric dimensions of all the connected networks under certain conditions. Moreover, we have calculated the fractional metric dimension of grid-like networks, called triangular and polaroid grids, with the aid of the aforementioned criteria. Moreover, we analyse the bounded and unboundedness of the fractional metric dimensions of the aforesaid networks with the help of 2D as well as 3D plots.