Classification of Upper Bound Sequences of Local Fractional Metric Dimension of Rotationally Symmetric Hexagonal Planar Networks

Kinkar Chandra Das,Fairouz Tchier,F. M. O. Tawfiq,Shahbaz Ali,Muhammad Khalid Mahmood

doi:10.1155/2021/6613033

Abstract

The term metric or distance of a graph plays a vital role in the study to check the structural properties of the networks such as complexity, modularity, centrality, accessibility, connectivity, robustness, clustering, and vulnerability. In particular, various metrics or distance-based dimensions of different kinds of networks are used to resolve the problems in different strata such as in security to find a suitable place for fixing sensors for security purposes. In the field of computer science, metric dimensions are most useful in aspects such as image processing, navigation, pattern recognition, and integer programming problem. Also, metric dimensions play a vital role in the field of chemical engineering, for example, the problem of drug discovery and the formation of different chemical compounds are resolved by means of some suitable metric dimension algorithm. In this paper, we take rotationally symmetric and hexagonal planar networks with all possible faces. We find the sequences of the local fractional metric dimensions of proposed rotationally symmetric and planar networks. Also, we discuss the boundedness of sequences of local fractional metric dimensions. Moreover, we summarize the sequences of local fractional metric dimension by means of their graphs.

Highlights

Academic Editor: Kinkar Chandra Das e term metric or distance of a graph plays a vital role in the study to check the structural properties of the networks such as complexity, modularity, centrality, accessibility, connectivity, robustness, clustering, and vulnerability
We find the sequences of the local fractional metric dimensions of proposed rotationally symmetric and planar networks
For a more accurate solution of integer programming problem (IPP), Currie and Oellermann introduced the concept of fractional metric dimension in [22]

Summary

Preliminaries

A resolving function θ′ of G is called a minimal resolving function if there exists u ∈ V(G) which is not a resolving function of G and if any other function θ: V(G) ⟶ [0, 1]. The θ(u) ≠ graph θG′(ius).defineefdraacstiodnimal fm(eGtr)ic dimension min􏼈|θ′|: θ′ is the minimal resolvingfunction of G}, where. E resolving function will be a local resoling function if θ′(R{s, t}) ≥ 1 and the fractional metric dimension becomes a local fractional metric dimension (LFMD) if we assume the pair of adjacent vertices only; it is denoted by Dimlf(G) [20, 28]. E LFMD of given graph in Figure 1 is Dimlf (G) ≤ (12/7). Since 14 is a minimal cardinal number, there must exist a minimal resolving function θ′ such that |θ′| ≤ |θ|.

Rotationally Symmetric Hexagonal Planar Networks

Conclusion