A study of some classes of fuzzy graphs with applications in decision-making

Abstract

Fuzzy graphs (FG) are crucial in various fields in both organic and human structures, such as simulation models in physiological, medical, and social structures. Since daily life prob lems are frequently ambiguous because of unclear and uncertain data, it is not easy for a specialist to model those problems using a FG. The complex fuzzy set (Com-FS) is repre sented by a complex-valued membership function that combines a traditional membership function (MF) with an additional component known as the phase term. The application of complex fuzzy theory can be beneficial in the field of mathematics because it gives the system more flexibility, consistency, and comparability than a fuzzy model. The Com-FS system is quite intricate because it allows for a large number of values for its MF. Unlike traditional fuzzy membership functions, which are limited to a range of [0, 1], the range for the Com-FS membership function includes the unit circle of the complex plane. This thesis aims to explore the properties of picture fuzzy graphs (PFG) as a way to analyze uncertain and inconsistent information related to real-valued problems, unlike FG, which does not provide mathematically reliable information. The objective is to reduce the lim itations of FG by introducing new definitions and results in PFG. The research will focus on the properties of PFG such as maximal product (Max-P), symmetric difference (Sym-D), rejection, and residue product (Res-P), as well as dτ(b) and tdτ(b) in PFG. Additionally, this study will examine the utilization of PFG in the fields of digital marketing and social media. A complex spherical fuzzy set (Com-SFS) is an extension of a spherical fuzzy set (SFS) and Com-FS that is used to deal with ambiguity more conclusively. This study aims to establish a complex spherical fuzzy graph (Com-SFG), which is an advancement from the complex Pythagorean fuzzy graph (Com-PyFG) due to its limitations in utilizing the com plex neutral membership function. The Com-SFG is considered more versatile compared to other fuzzy models, such as complex fuzzy models, complex intuitionistic fuzzy models, and complex Pythagorean fuzzy models, as it incorporates three aspects: complex membership, neutral membership, and non-membership functions. The Com-FS uses complex numbers to represent membership in a set, providing a mathematical structure for this purpose. Re cently, this mathematical technique has gained popularity for combining different attributes. xi By utilizing this technique, we introduce powerful approaches for Com-FG properties. The purpose of this research is to determine the degree of a vertex (dτ(b)), size, order, and total degree of vertex (tdτ(b)) of Com-FG. We do it over fundamental operations like join, com plement, and union in Com-FG. We investigate the application of Com-FG. Further, we will discuss the motivation for Com-SFG. We define dτ(b), order, size, and dτ(b) of Com-SFG. We describe some basic operations, such as join, union, and complement, in Com-SFG. On Com-SFG, this research study introduces several operations, including composition, strong product, Cartesian product, and semi-strong product. Moreover, we present the Com-SFG application, which ensures the ability to deal with problems in three directions