# Energy of Pythagorean Fuzzy Graphs with Applications
## 1 Introduction  
[PERSON] recently [1; 2] introduced the concept of the Pythagorean fuzzy set (PFS) as a generalization of the intuitionistic fuzzy set (IFS) [3], to manage the complex impreciseness and uncertainty in practical decision-making problems. The prominent characteristic of the Pythagorean fuzzy model is to relax the condition that the sum of its membership degree and non-membership degree is no greater than one with the square sum of its membership degree and non-membership degree no greater than one. After the inception of PFS by [PERSON] [2], [PERSON] and [PERSON] [4] presented the mathematical form of the PFS and introduced the concept of the Pythagorean fuzzy number (PFN). Meanwhile, they presented a series of basic operational laws of PFNs and proposed the Pythagorean fuzzy aggregation operators. PFS, a novel class of the non-standard fuzzy set, has a wide range of applications in different fields, such as medical diagnosis [5], Internet stock investment [6], the service quality of domestic airlines [4] and the governor selection of the Asian Infrastructure Investment Bank [7]. Graph representations are generally used for dealing with structural information, in different domains such as operations research, networks, systems analysis, pattern recognition, economics and image interpretation. [PERSON] [8] introduced the notion of the energy of a graph in chemistry, because of its relevance to the total \(\pi\)-electron energy of certain molecules and found upper and lower bounds for the energy of graphs [9]. In chemistry, the energy of a given molecular graph is interesting because of its relation to the total \(\pi\)-electron energy of the molecule represented by that graph. A graph with all isolated vertices \(K_{n}^{c}\) has zero energy, while the complete graph \(K_{n}\) with \(n\) vertices has energy \(2(n-1)\). Later, [PERSON] and [PERSON] [10] defined the Laplacian energy of a graph as the sum of the absolute values of the differences of the average vertex degree of \(G\) to the Laplacian eigenvalues of \(G\). When there is obscureness in the description of the objects, or in their relations, or in both, the fuzzy graph model is put forward naturally. The concept of fuzzy graphs was initiated by [PERSON] [11], based on [PERSON]'s fuzzy relations [12]. [PERSON] [13] discussed the concept of the fuzzy graph and developed its structure. The energy of a fuzzy graph was investigated in [14] byAnjali and [PERSON]. The Laplacian energy of a fuzzy graph was defined by [PERSON] and [PERSON] [15]. [PERSON] and [PERSON] [16] generalized the concept of a fuzzy graph to an intuitionistic fuzzy graph (IFG). Later, IFGs were discussed by [PERSON] and [PERSON] [17]. [PERSON] et al. [18] defined the energy of IFGs as an extension of [14]. [PERSON] and [PERSON] [19] generalized the concept of the Laplacian energy of a fuzzy graph to the Laplacian energy of an IFG. [PERSON] et al. [20; 21; 22; 23; 24; 25; 26] put forward many new concepts concerning the extended structures of fuzzy graphs and provided their pertinent applications in decision-making. Recently, [PERSON] et al. [27] proposed the concept of Pythagorean fuzzy graphs (PFGs), a generalization of the notion of [PERSON] and [PERSON] IFGs [17], along with its applications in decision-making. The Pythagorean fuzzy model is more flexible and practical than fuzzy and intuitionistic fuzzy models. Therefore, in this research study, we introduce certain novel concepts, including the energy and Laplacian energy of PFGs, as well as the energy and Laplacian energy of Pythagorean fuzzy digraphs (PFDGs). We illustrate these concepts with examples. We investigate some of their interesting properties. In particular, we solve decision-making problems concerning the design of a satellite communication system and the evaluation of the schemes of reservoir operation to illustrate the applicability and effectiveness of our proposed notions. The paper is structured as follows: Section 2 proposes the concept of the energy of a PFG and investigates its properties.