Spherical Fuzzy Graphs with Application to Decision-Making

Mathematics, 2019

The picture fuzzy set is an efficient mathematical model to deal with uncertain real life problems, in which a intuitionistic fuzzy set may fail to reveal satisfactory results. Picture fuzzy set is an extension of the classical fuzzy set and intuitionistic fuzzy set. It can work very efficiently in uncertain scenarios which involve more answers to these type: yes, no, abstain and refusal. In this paper, we introduce the idea of the picture fuzzy graph based on the picture fuzzy relation. Some types of picture fuzzy graph such as a regular picture fuzzy graph, strong picture fuzzy graph, complete picture fuzzy graph, and complement picture fuzzy graph are introduced and some properties are also described. The idea of an isomorphic picture fuzzy graph is also introduced in this paper. We also define six operations such as Cartesian product, composition, join, direct product, lexicographic and strong product on picture fuzzy graph. Finally, we describe the utility of the picture fuzzy ...

Mathematical Problems in Engineering

The concept of fuzzy graph (FG) and its generalized forms has been developed to cope with several real-life problems having some sort of imprecision like networking problems, decision making, shortest path problems, and so on. This paper is based on some developments in generalization of FG theory to deal with situation where imprecision is characterized by four types of membership grades. A novel concept of T-spherical fuzzy graph (TSFG) is proposed as a common generalization of FG, intuitionistic fuzzy graph (IFG), and picture fuzzy graph (PFG) based on the recently introduced concept of T-spherical fuzzy set (TSFS). The significance and novelty of proposed concept is elaborated with the help of some examples, graphical analysis, and results. Some graph theoretic terms are defined and their properties are studied. Specially, the famous Dijkstra algorithm is proposed in the environment of TSFGs and is applied to solve a shortest path problem. The comparative analysis of the propose...

Journal of Intelligent & Fuzzy Systems, 2019

In this paper, we introduce spherical fuzzy sets (SFSs) which is an advanced tool of the fuzzy sets, intuitionistic fuzzy sets and picture fuzzy sets. We investigate the basic properties of SFSs and compare the idea SFSs with picture fuzzy sets. The spaces of spherical membership grades and picture membership grades are examined by graphically. Some useful operation such as spherical fuzzy t'-norms and spherical fuzzy t'-conorms are introduced. Also introduced spherical fuzzy negator and some classifications of spherical fuzzy t'-norms and spherical fuzzy t-conorms which are useful for development the aggregation operator to aggregate the spherical fuzzy information. Finally, a food circulation center evaluation problem is given as a practical application to demonstrate the usage and applicability of the proposed ranking approach.

Handbook of Research on Advanced Applications of Graph Theory in Modern Society

In this chapter, the authors introduce some basic definitions related to fuzzy graphs like directed and undirected fuzzy graph, walk, path and circuit of a fuzzy graph, complete and strong fuzzy graph, bipartite fuzzy graph, degree of a vertex in fuzzy graphs, fuzzy subgraph, etc. These concepts are illustrated with some examples. The recently developed concepts like fuzzy planar graphs are discussed where the crossing of two edges are considered. Finally, the concepts of fuzzy threshold graphs and fuzzy competitions graphs are also given as a generalization of threshold and competition graphs.

Journal of Mathematics

The purpose of the study is to explore graph theory based on cubic Pythagorean fuzzy sets. The concept of cubic Pythagorean fuzzy graphs (CuPFGs) is introduced in this research work. In addition, we define certain fundamental operations on CuPFGs including semistrong product, lexicographical product, and symmetric difference of two CuPFGs and demonstrate some of their key characteristics. Meanwhile, to investigate the preference of decision makers, cubic Pythagorean fuzzy preference relation is defined. Moreover, through a practical example, the applicability of our proposed work in multicriteria decision-making is described, and to clarify the organization of the proposed method, a frame diagram is presented.

IEEE Access

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.

SpringerPlus, 2016

Background Graph theory have applications in many areas of computer science including data mining, image segmentation, clustering, image capturing, networking. A graph structure, introduced by Sampathkumar (2006), is a generalization of undirected graph which is quite useful in studying some structures including graphs, signed graphs, graphs in which every edge is labeled or colored. A graph structure helps to study the various relations and the corresponding edges simultaneously. A fuzzy set (Zadeh 1965) is an important mathematical structure to represent a collection of objects whose boundary is vague. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional models used in engineering and science. Nowadays fuzzy sets are playing a substantial role in chemistry, economics, computer science, engineering, medicine and decision making problems. In 1998, Zhang (1998) generalized the idea of a fuzzy set and gave the concept of bipolar fuzzy set on a given set X as a map which associates each element of X to a real number in the interval [−1, 1]. In 2014, Chen et al. (2014) introduced the idea of m-polar fuzzy sets as an extension of bipolar fuzzy sets and showed that bipolar fuzzy sets and 2-polar fuzzy sets are cryptomorphic mathematical notions and that we can obtain concisely one from the corresponding one in Chen et al. (2014). The idea behind this is that "multipolar information" (not just bipolar information which corresponds to two-valued logic) exists because data for a real world problem are sometimes from n agents (n ≥ 2). For example, the exact degree of telecommunication safety of mankind is a point in [0, 1] n (n ≈ 7 × 10 9) because different person has been monitored

2020

In this paper picture fuzzy graph has shown more advantage in handling vagueness and uncertainty than fuzzy set. We have applied concept of picture fuzzy set to fuzzy set. Picture fuzzy graph is generalization of fuzzy graph. The concept of picture fuzzy graph has discussed here. In this paper complement and Ring sum operation of picture fuzzy graphs has discussed isomorphic property of picture fuzzy graphs. We have used picture Fuzzy graph play important role in representation of many uncertain decision making problem in daily life since Picture fuzzy set is an efficient mathematical model to deal with uncertain real life problems, e.g. number theory, coding theory, cryptography, computer science, operation research, also we are using certain concepts of bipolar fuzzy directed hypergraphs Finally we have proved the Ring sum of two picture fuzzy graphs is also a picture fuzzy graph.

Mathematical and Computational Applications

The purpose of this research study is to present some new operations, including rejection, symmetric difference, residue product, and maximal product of Pythagorean fuzzy graphs (PFGs), and to explore some of their properties. This research article introduces certain notions, including intuitionistic fuzzy graphs of 3-type (IFGs3T), intuitionistic fuzzy graphs of 4-type (IFGs4T), and intuitionistic fuzzy graphs of n-type (IFGsnT), and proves that every IFG(n−1)T is an IFGnT (for n ≥ 2). Moreover, this study discusses the application of Pythagorean fuzzy graphs in decision making.

Complex & Intelligent Systems

Graph theory plays crucial role in structuring many real-world problems including, medical sciences, control theory, expert systems and network security. Product in graphs, an operation that consider two graphs and produce a new graph by simple or complex changes, has wide range of applications in games theory, automata theory, structural mechanics and networking system. An intuitionistic fuzzy model is used to handle the vagueness and uncertainty in network problems. A Pythagorean fuzzy model is a powerful tool for describing vagueness and uncertainty more accurately as compared to intuitionistic fuzzy model. The objective of this paper is to apply the concept of Pythagorean fuzzy sets to graphs and then combine two Pythagorean fuzzy graphs (PFGs) using two new graph products namely, maximal product and the residue product. This research paper investigates the regularity for these products. Moreover, it discusses some eminent properties such as strongness, connectedness and completeness. Further, it proposes some necessary and sufficient conditions for G 1 * G 2 and G 1 • G 2 to be regular. Finally, decision-making problems concerning evaluation of best company for investment and alliance partner selection of a software company are solved to better understand PFGs.