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Pythagorean fuzzy graph, a broadly used extension of fuzzy and intuitionistic fuzzy graph, is helpful in representing structural relationships between several objects where the relation between these objects is vague, while the Dombi operators with operational parameters have excellent flexibility. Utilizing these two concepts, this research paper proposes the novel concept of Pythagorean Dombi fuzzy graphs (PDFGs). Basically, graph terminology is employed for introducing Pythagorean fuzzy analogs of various fundamental graphical ideas using Dombi operator. Further, under Pythagorean Dombi fuzzy environment, regular, totally regular, strongly regular and biregular graphs are defined with appropriate illustration and some of their crucial properties are examined. Meanwhile, the notion of edge regularity of PDFG is also initiated with substantial characteristics. Finally, a numerical example related to evaluation of appropriate ETL software for a business intelligence project is presented to better understand PDFGs.

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 $${\mathscr {G}}_{1}*{\mathscr {G}}_{2}$$ and $${\mathscr {G}}_{1}\cdot {\mathscr {G}}_{2}$$ 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.

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.

Graph theory plays a substantial role in structuring and designing many problems. A number of structural designs with crossings can be found in real world scenarios. To model the vagueness and uncertainty in graphical network problems, many extensions of graph theoretical ideas are introduced. To deal with such uncertain situations, the present paper proposes the concept of Pythagorean fuzzy multigraphs and Pythagorean fuzzy planar graphs with some of their eminent characteristics by investigating Pythagorean fuzzy planarity value with strong, weak and considerable edges. A close association is developed between Pythagorean fuzzy planar and dual graphs. This paper also includes a brief discussion on non-planar Pythagorean fuzzy graphs and explores the concepts of isomorphism, weak isomorphism and co-weak isomorphism for Pythagorean fuzzy planar graphs. Moreover, it presents a problem that shows applicability of the proposed concept.