Multicriteria decision-making based on the degree and distance-based indices of fuzzy graphs

The topological indices are found to be interesting in fuzzy graph theory. A fuzzy topological index is a value that depends on the fuzzy edge membership, vertex membership, fuzzy distance or fuzzy degree, etc. The modeling of various fuzzy models such as connectivity models and multi-criteria decision-making problems can be investigated using the fuzzy topological indices. In this paper, our aim is to investigate the fuzzy topological indices based on both degrees of vertices and distances between the vertices, namely, the fuzzy degree-distance index and the fuzzy Gutman index for fuzzy graphs. In particular, we discuss the behavior of these indices under certain graph operations. The lower and upper bounds of these fuzzy indices for fuzzy graphs and fuzzy regular graphs are presented. The fuzzy second Zagreb coindex is defined for fuzzy graphs. The lower and upper bounds of the fuzzy Gutman index in terms of fuzzy second Zagreb and Zagreb coindex are established. The average degree-distance index for the fuzzy graphs is defined. Based on this average index, vertices are classified into fuzzy degree-distance enhancing vertices, fuzzy degree-distance reducing vertices, and fuzzy degree-distance neutral vertices. An algorithm to find the fuzzy degree-distance index for the model of multi-criteria decision making problem is presented. Finally, we implement our model of fuzzy degree-distance index is implemented to an Urban Public Transportation Problem for finding the best place for a bus stop. The obtained results are compared with already existing models.