Abstract
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 proposed concept and existing theory is made. In addition, the advantages of the proposed work are discussed over the existing tools.
Highlights
In the past decades, the development of graph theory, the fuzzy graph (FG) theory, and its applications in numerous scientific subjects indicates its significance. e addition of FGs in graph theory is of worth as it increases the viability of graph theory
From application point of view, FGs have been widely utilized in practical problems, for example, reference [1] provided a list of possible regions handled by FGs and fuzzy hypergraphs, reference [2] modelled some traffic problems using FGs, reference [3] utilized FGs in optimization of networks, reference [4] is based on application of telecommunication system in FGs, and reference [5] applied FGs in fuzzy neural networks. e theory of FGs has been initiated in [6] but briefly elaborated in [7] by Rosenfield after the remarkable work of Zadeh [8] on fuzzy sets (FSs)
The concept of T-spherical fuzzy graph (TSFG) is introduced based on the novel theory of T-spherical fuzzy set (TSFS)
Summary
The development of graph theory, the fuzzy graph (FG) theory, and its applications in numerous scientific subjects indicates its significance. e addition of FGs in graph theory is of worth as it increases the viability of graph theory. Due to this formation of PFSs, one is unable to assign the values to these membership, abstinence, and non-membership functions by their own choice Keeping this issue in mind, Mahmood et al [32] proposed the concept of spherical fuzzy sets (SFSs) and T-spherical fuzzy sets (TSFSs), which improves the construction of PFS and does not have limitations at all. To discuss the diversity and significance of TSFGs, a shortest path problem in the environment of TSFSs and TSFGs is studied.
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