Group decision-making framework under linguistic q-rung orthopair fuzzy Einstein models

The q-rung orthopair fuzzy sets dynamically change the range of indication of decision knowledge by adjusting a parameter q from decision makers, where \(q \ge 1\), and outperform the conventional intuitionistic fuzzy sets and Pythagorean fuzzy sets. Linguistic q-rung orthopair fuzzy sets (Lq-ROFSs), a qualitative type of q-rung orthopair fuzzy sets, are characterized by a degree of linguistic membership and a degree of linguistic non-membership to reflect the qualitative preferred and non-preferred judgments of decision makers. Einstein operator is a powerful alternative to the algebraic operators and has flexible nature with its operational laws and fuzzy graphs perform well when expressing correlations between attributes via edges between vertices in fuzzy information systems, which makes it possible for addressing correlational multi-attribute decision-making (MADM) problems. Inspired by the idea of Lq-ROFS and taking the advantage of the flexible nature of Einstein operator, in this paper, we aim to introduce a new class of fuzzy graphs, namely, linguistic q-rung orthopair fuzzy graphs (Lq-ROFGs) and further explore efficient approaches to complicated MAGDM situations. Following the above motivation, we propose the new concepts, including product-connectivity energy, generalized product-connectivity energy, Laplacian energy and signless Laplacian energy and discuss several of its desirable properties in the background of Lq-ROFGs based on Einstein operator. Moreover, product-connectivity energy, generalized product-connectivity energy, Laplacian energy and signless Laplacian energy of linguistic q-rung orthopair fuzzy digraphs (Lq-ROFDGs) are presented. In addition, we present a graph-based MAGDM approach with linguistic q-rung orthopair fuzzy information based on Einstein operator. Finally, an illustrative example related to the selection of mobile payment platform is given to show the validity of the proposed decision-making method. For the sake of the novelty of the proposed approach, comparison analysis is conducted and superiorities in contrast with other methodologies are illustrated.