Abstract
The q-rung orthopair fuzzy graph is an extension of intuitionistic fuzzy graph and Pythagorean fuzzy graph. In this paper, the degree and total degree of a vertex in q-rung orthopair fuzzy graphs are firstly defined. Then, some product operations on q-rung orthopair fuzzy graphs, including direct product, Cartesian product, semi-strong product, strong product, and lexicographic product, are defined. Furthermore, some theorems about the degree and total degree under these product operations are put forward and elaborated with several examples. In particular, these theorems improve the similar results in single-valued neutrosophic graphs and Pythagorean fuzzy graphs.
Highlights
In 2017, Yager proposed the concept of q-rung orthopair fuzzy sets (q-ROFSs) [1], which is a generalization of intuitionistic fuzzy sets (IFSs) [2] and Pythagorean fuzzy sets (PFSs) [3,4]
The implications of the degree and total degree of a vertex in q-rung orthopair fuzzy graph (q-ROFG) are illustrated by the example of road network
The degree and total degree of a vertex help one understand the properties of the product operations on q-ROFGs
Summary
In 2017, Yager proposed the concept of q-rung orthopair fuzzy sets (q-ROFSs) [1], which is a generalization of intuitionistic fuzzy sets (IFSs) [2] and Pythagorean fuzzy sets (PFSs) [3,4]. Compared with IFSs and PFSs, q-ROFSs provide decision-makers more elasticity to voice opinions with respect to membership grades of an element. Akram et al [24] investigated some product operations of PFGs and the degree and total degree of a vertex in PFGs. the product operations on q-ROFGs have not been researched yet, so we will pay our attention to this subject in this paper. We have found that in SVNGs and PFGs, the results about the degree and total degree under some product operations fail to work in some cases. To improve these results, we introduced the number of adjacent vertices and obtained some more general theorems.
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