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.

的q -rung orthopair模糊集通过调整参数动态地改变的决策知识指示的范围q从决策者，其中\（Q \ GE 1 \） ，和优于常规直觉模糊集和毕达哥拉斯模糊集合。语言q -rung orthopair模糊集合（L q -ROFSs），质型的q阶邻对模糊集的特征是一定程度的语言隶属度和一定程度的语言非隶属度，以反映决策者的定性偏好和非偏好判断。爱因斯坦算子是代数算子的强大替代品，并且具有灵活的性质，其操作定律和模糊图在通过模糊信息系统中的顶点之间的边表示属性之间的相关性时表现良好，这使得解决相关的多属性决策成为可能（MADM）问题。受到L q -ROFS的启发，并利用爱因斯坦算子的灵活性质，本文旨在引入一类新的模糊图，即语言q-阶邻对模糊图（Lq -ROFGs），并进一步探索解决复杂MAGDM情况的有效方法。遵循上述动机，我们提出了新概念，包括乘积连接能，广义乘积连接能，拉普拉斯能和无符号拉普拉斯能，并在基于爱因斯坦算子的L q -ROFGs背景下讨论了其一些理想的性质。此外，还给出了语言q-阶邻对对有向图（L q -ROFDGs）的乘积连通能，广义乘积连通能，拉普拉斯能和无符号拉普拉斯能。此外，我们提出了一种基于图的MAGDM方法，该方法具有语言学上的q爱因斯坦算子的双邻位对对模糊信息。最后，给出了一个与移动支付平台选择有关的说明性例子，以说明所提出的决策方法的有效性。为了使所提出的方法新颖，进行了比较分析，并说明了与其他方法相比的优越性。