Abstract
Graphs play a pivotal role in structuring real-world scenarios such as network security and expert systems. Numerous extensions of graph theoretical conceptions have been established for modeling uncertainty in graphical network situations. The Pythagorean Dombi fuzzy graph (PDFG), a generalization of the fuzzy Dombi graph (FDG), is very useful in representing vague relations between several objects, whereas the operational parameter has a flexible nature in decision-making problems. The main objective of this research study is to expand the area of discussion on PDFGs by establishing fruitful results and notions related to operations such as the direct product, Cartesian product, semi-strong product, strong product, and composition on PDFGs. Certain concepts, including the degree of vertices and total degree, are discussed as its modifications. Meanwhile, these outcomes are considered on PDFGs maintaining the strongness property. At the end, an algorithm for Pythagorean Dombi fuzzy multi-criteria decision-making is given, and a numerical example based on the selection of a leading textile industry is put forward to clarify the suitability of the proposed approach.
Highlights
In the last several years, many operators were established that occurred in various monographs with regard to fuzzy logic; min-max, Frank, Einstein, product, Hamacher, and Dombi operators
It is seen that restrictions 0 ≤ μ ≤ 1, 0 ≤ μ + ν ≤ 1 on fuzzy graph (FG) and IFG, respectively, confine the area of these graphs to describe uncertain information that appears in the real world
We showed that the Cartesian product, strong product, semi-strong product, and the composition of two Pythagorean Dombi fuzzy graph (PDFG) were not PDFGs
Summary
In the last several years, many operators were established that occurred in various monographs with regard to fuzzy logic; min-max, Frank, Einstein, product, Hamacher, and Dombi operators. As an extensive range of applications, such as database theory, optimization of networks, and decision-making are covered by means of graphs, on this basis, Naz et al [29] presented the notion of Pythagorean fuzzy graphs (PFGs) by considering min and max operators. Since FG can model and structure decision-making situations with vagueness, a very insufficient attempt has been made to utilize the Dombi operator in graph theory. On this base, Ashraf et al [48] provided the notion of the Dombi fuzzy graph (DFG). If s1t1 ∈ E1 and s2t2 ∈ E2, the membership grade is: it is concluded that G1 × G2 is not a strong PDFG of G1 × G2, a contradiction
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