Spherical Fuzzy Graph Application in Traffic

Abstract

The applications of spherical fuzzy graphs are covered in this article. Real-world issues are addressed with the spherical fuzzy graph. The purpose of the graph is to use spherical fuzzy vertex colouring to indicate the unintentional zone in the traffic flows. The applications, on the other hand, display information regarding the movement and intersection of the cars. In order to colour the nodes in a network that generates the chromatic number, this paper\'s goal is to highlight the intersection areas on road maps.

Introduction

I. INTRODUCTION

A branch of mathematics is called graph theory. Sylvester introduced the word "graph." Graphs are more than just bar and line graphs; they also include a pair of vertices and the edges that connect them. Numerous branches of science and technology employ graph theory. The concept of graph colouring is used in many contexts, including scheduling, aviation, traffic signals, etc. In the 18th century, Swiss mathematician Leonhard Euler introduced the fundamental concepts of graphs. It plays a significant part in a real-life dilemma. L.A. Zadeh independently published his first work in fuzzy set in 1965. It is a helpful tool to describe circumstances where the data are ambiguous. The scenarios in which the items belong to a set are handled by fuzzy sets to some extent by their attributes. Only memberships were used when Zadeh invented the fuzzy set. Atanassov expanded on the research of fuzzy sets by introducing non-memberships and intuitionist fuzzy sets in 1982. He also suggested applications in decision-making, system theory, and other fields. Atanassov fuzzy set and Atanassov intuitionist fuzzy set are other names for intuitionist fuzzy set. Pythagorean fuzzy set is a fresh expansion of intuitionist fuzzy set (IFS) (PFS). Yager (2013) introduced the Pythagorean fuzzy set to deal with the complicated uncertainty. He then developed the idea of an intervalued Pythagorean fuzzy set from the Pythagorean fuzzy set. Later, Smarandache developed the neutrosophic fuzzy set (NFS), which measures the degree of ambiguity (1995). It is a generalisation of the single-valued neutrosophic set and the intuitionist fuzzy set.

Introducing now the intuitive fuzzy set and fuzzy set's advanced tool. As an expansion of the Pythagorean fuzzy set, Gundogdu (2018) and Kahraman initially proposed the spherical fuzzy set (SFS). It was a flexible model that addressed a variety of real-world scenarios. Later, Rosenfeld's fuzzy graph from 1975 takes on this shape. Atanassov (1999) further developed fuzzy graph, which was an expansion of fuzzy set theory. Munoz et al. posed the question about the fuzzy graph's chromatic number first. Parvathi and Karunambigai (2006) give the definition of intuitionist fuzzy graph and its properties. The extended idea of Pythagorean fuzzy set to Pythagorean fuzzy graph was initiated by Muhammad Akram et.al (2019). (2019). The concept of spherical fuzzy graphs was introduced by Akram et al in 2020. The interval condition gives a truthness, falseness, and indeterminacy. Spherical fuzzy graphs are easier to utilise than image fuzzy graphs for a variety of real-world circumstances. The four colour problem, which was the most well-known and fascinating topic in graph theory, had its beginnings with Francis Guthrie (1852). Eslahchi and Onagh (2006) also developed the fuzzy graph's colouring. In Myna (2015), the use of fuzzy graphs in traffic was considered. In this paper, we simply touch on the use of spherical fuzzy vertex colouring for traffic control. This spherical fuzzy graph shows how traffic lights are used to direct vehicles and how they cross, with the unintentional zone denoted. We constructed the spherical fuzzy graph and coloured nodes using the motion of passing automobiles. Finding a graph's chromatic number, also known as the minimal colouring, serves as its conclusion. Additionally, utilising traffic flows, we gave a numerical example of a spherical fuzzy graph.

IV. GRAPHICALLY ILLUSTRATE A TRAFFIC FLOW IN A SPHERE

This article uses a spherical fuzzy graph to depict a vehicle flow graph. We depicted a vehicle flow that encountered an accident at a traffic light system (Fig 4.1). Vehicles are referred to as vertices or nodes, and the path they take as they change directions is referred to as an edge. The directions are marked as A, B, C, and D. If there are no vehicles on the left sides in either direction, then there is no intersection. When the signals are active, a collision between the vehicles occurs. This strategy is illustrated by the example below: Each arrow denotes a vehicle in motion. Vehicle density in each path is not always the same. The alternative traffic directions are depicted in figure 4.1, and the two left turns in that figure have less vehicles and do not impede traffic flow. Each vehicle flow represents the fuzzy spherical edge. In Fig. 4.1, A, B, and C were involved in an accident. In this scene, vehicles from A were travelling in the direction of path D, while those from B crossed over A. The intersection is caused by a collision between the three vertices, which is known as an unintentional zone. Figure 4 is used to calculate the chromatic number. 2. Because it intersects at one point, the vertices v? and v? are referred to be neighbouring.

V. ACKNOWLEDGEMENT

Thank you to M. RAMYA, Assistant Professor in Department of Mathematics at Dr.SNS Rajalakshmi College of Arts and Science who has supported this research.

Conclusion

This article covered the use of spherical fuzzy graphs in traffic control applications. The vehicles are designated as membership values and as a graph vertex. Vehicle travelling in both non-membership values and as an edge from one direction to another. Here, we merely colour a spherical fuzzy graph\'s vertex edges and look at how nodes are coloured. The vertices that are known to be chromatic numbers are minimally coloured as a result of this study.

References

[1] Rupkumar Mahapatra¹, Sovan Samanta ², and Madhumangal Pal, Edge Colouring of Neutrosophic Graphs and Its Application in Detection of Phishing Website, Published 8 July 2022. [2] V.N.Srinivasa Rao Repalle, Lateram Zawuga Hordofa, and Mamo Abebe Ashebo, Chromatic Polynomial of Intuitionistic Fuzzy Graphs Using (?, ?)- Levels, Published 28 June 2022 [3] S. M. Sudha, K. Akalyadevi a), K. Preethi Sowndarya, Application of Spherical Fuzzy Graph in Traffic, Cite as: AIP Conference Proceedings 2393, 020216 (2022); https://doi.org/10.1063/5.0074402 Published Online: 19 May 2022. [4] Abdul.Muneera¹,Dr.T.Nageswara Rao², Dr.R.V.N.Srinivasa Rao³, Dr.J.Venkateswara Rao?, Application Of Fuzzy Graph Theory Portrayed In Various Fields. Posted Date: March 19th, 2021. [5] Wei Gao, Weifan Wang, Overview on Fuzzy Fractional Coloring, International Journal of Cognitive Computing in Engineering 2 (2021) 196-201.

Copyright

Copyright © 2022 Kaviya V, Ramya. M. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.