In recent years, the concept of domination has been the backbone of research activities in graph theory. The application of graphic domination has become widespread in different areas to solve human-life issues, including social media theories, radio channels, commuter train transportation, earth measurement, internet transportation systems, and pharmacy. The purpose of this paper was to generalize the idea of bondage set (BS) and non-bondage set (NBS), bondage number $\alpha \left(G\right)$, and non-bondage number ${\alpha }_k\left(G\right)$, respectively, in the intuitionistic fuzzy graph (IFG). The BS is based on a strong arc (SA) in the fuzzy graph (FG). In this research, a new definition of SA in connection with the strength of connectivity in IFGs was applied. Additionally, the BS, $\alpha \left(G\right)$, NBS, and ${\alpha }_k\left(G\right)$ concepts were presented in IFGs. Three different examples were described to show the informative development procedure by applying the idea to IFGs. Considering the examples, some results were developed. Also, the applications were utilized in water supply systems. The present study was conducted to make daily life more useful and productive.

Fuzzy graphs (FGs) can play a useful role in natural and human-made structures, including process dynamics in physical, biological, and social systems. Since issues in everyday life are often uncertain due to inconsistent and ambiguous information, it is extremely difficult for an expert to model those difficulties using an FG. Indeterminate and inconsistent information related to real-valued problems can be studied through a picture of the fuzzy graph (PFG), while the FG does not provide mathematically acceptable information. In this regard, we are interested in reducing the limitations of FGs by introducing some new definitions and results for the PFG. This paper aims to describe and explore a few properties of PFGs, including the maximal product (MP), symmetric difference (SD), rejection (RJ), and residue product (RP). Furthermore, we also discuss the degree and total degree of nodes in a PFG. This study also demonstrates the application of a PFG in digital marketing and social networking.

A complex picture fuzzy set (Com-PFS) is a motivating tool for more precisely interpreting fuzzy notions. Recently, all extensions of complex fuzzy graphs (Com-FGs) have become a growing research topic as they handle ambiguous situations more explicitly than complex intuitionistic fuzzy graphs (Comp-IFG) and picture fuzzy graphs (PFG). The primary goal of this study is to demonstrate the foundation of Com-PFG due to the drawback of the complex neutral membership function in Com-IFG. This paper introduces the concept of Com-PFGs and explains many of the approaches to their development. A Com-PFG has three complex membership functions. We define the order, size, degree of vertex, and total degree of vertex in Com-PFGs. We elaborate on primary operations including: complement, join, union, Cartesian product, direct product, and composition of Com-PFG. Finally, we discuss its use in decision-making problems.

A complex fuzzy set (CFS) is described by a complex-valued truth membership function, which is a combination of a standard true membership function plus a phase term. In this paper, we extend the idea of a fuzzy graph (FG) to a complex fuzzy graph (CFG). The CFS complexity arises from the variety of values that its membership function can attain. In contrast to a standard fuzzy membership function, its range is expanded to the complex plane’s unit circle rather than [0,1]. As a result, the CFS provides a mathematical structure for representing membership in a set in terms of complex numbers. In recent times, a mathematical technique has been a popular way to combine several features. Using the preceding mathematical technique, we introduce strong approaches that are properties of CFG. We define the order and size of CFG. We discuss the degree of vertex and the total degree of vertex of CFG. We describe basic operations, including union, join, and the complement of CFG. We show new maximal product and symmetric difference operations on CFG, along with examples and theorems that go along with them. Lastly, at the base of a complex fuzzy graph, we show the application that would be important for measuring the symmetry or asymmetry of acquaintanceship levels of social disease: COVID-19.

A complex spherical fuzzy set (CSFS) is a generalization of a spherical fuzzy set (CFS). CSFS handles vagueness more explicitly, and its range is expanded from the real subset to the complex with unit disc. The major goal of this research is to present the foundation of a complex spherical fuzzy graph (CSFG) due to the limitation of the complex neutral membership function in a complex Pythagorean fuzzy graph (CPFG). Complex spherical fuzzy models have more flexibility as compared to complex fuzzy models, complex intuitionistic fuzzy models, and complex Pythagorean fuzzy models due to their coverage in three directions: complex membership functions, neutral membership functions, and complex non-membership functions. Firstly, we present the motivation for CSFG. Furthermore, we define the order, degree of a vertex, size, and total degree of a vertex of CSFG. We elaborate on primary operations, including complement, join, and the union of CSFG. This research study introduces some operations, namely, strong product, composition, Cartesian product, and semi-strong product, on CSFG. Moreover, we present the application of CSFG, which ensures the ability to deal with problems in three directions.

A complex fuzzy set (CFS) is a generalization of a fuzzy set (FS) in which a limit of degrees occurs on the complex plane with unit disc. The averaging operators are a key part of turning all the data into one value. Dombi operators have exceptional flexibility with operating factors, and they are particularly efficient in decision-making problems. In this paper, we establish a complex dombi fuzzy graph (CDFG). We implement dombi operators on CFSs to extend graph nomenclature. We define the complement of CDFG with an example. The idea of self-complementary in CDFG is discussed. The concepts of homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism of two CDFGs are discussed. We define regular and entirely regular graphs with sufficient elaboration and examine their key properties. Furthermore, significant characteristics are used to explain the edge regularity of CDFG. Lastly, we establish an application of CDFG in decision making problems.