# Energy of Pythagorean Fuzzy Graphs with Applications
## 5 Applications of the Energy of PFGs in Decision-Making
### Designing of a Satellite Communication System  
Communication is closely related to social development. In particular, satellite communication [42] has wide coverage area for communication, without geographical restrictions, and is less susceptible to the impact of land disasters. There is a unique advantage of satellite communication in many application areas, such as remote areas, islands, mountains, voyage aircraft and oceangoing vessels. Therefore, satellite communication not only effectively supplements the lack of other means of communication, but also has an irreplaceable role as the primary method of communication in mass media, especially the military. In modern warfare, it is very important to quantify the quality of the communication service. Suppose that a communication joint department in China plans to design a satellite communication system. Thus, a new test methodology for the synthetic communication system needs to be investigated, in order to provide evidence for building satellite Earth stations in the future. According to the expeditions, there are four possible testing venues \(z_{i}\)\((i=1,2,3,4)\) to choose from (Xichang \(z_{1}\), Chengdu \(z_{2}\), Nanjing \(z_{3}\) and Lhasa \(z_{4}\)). However, due to resources, funds, and other factors, only the most suitable of these would be selected. A decision-making group composed of six experts \(e_{k}\)\((k=1,2,\ldots,6)\) provides the judgments with six individual Pythagorean fuzzy preference relations (PFPRs) [27]\(R_{k}=(r_{ij}^{(k)})_{4\times 4}\)\((k=1,2,\ldots,6)\) as follows:  
\[R_{1}=\left[\begin{array}{cccc}(0.5,0.5)&(0.6,0.5)&(0.3,0.6)&(0.7,0.5)\\ (0.5,0.6)&(0.5,0.5)&(0.7,0.6)&(0.8,0.5)\\ (0.6,0.3)&(0.6,0.7)&(0.5  
The PFDGs \(\mathcal{D}_{k}\) corresponding to PFPRs given in matrices \(R_{k}\)\((k=1,2,\ldots,6)\) are shown in Figure 6.  
The energy of each PFDG is calculated as:  
\[E(\mathcal{D}_{1}) =(3.5008,3.5008),E(\mathcal{D}_{2})=(3.3021,3.3021),E(\mathcal{D}_ {3})=(3.0706,3.0706),\] \[E(\mathcal{D}_{4}) =(3.1675,3.1675),E(\mathcal{D}_{5})=(3.1860,3.1860),E(\mathcal{D}_ {6})=(3.0656,3.0656).\]  
Then, the weight of each expert can be calculated as:  
\[w_{k}=((w_{n})_{k},(w_{v})_{k})=\left(\frac{E((\mathcal{D}_{\mu})_{k})}{\sum \limits_{l=1}^{s}E((\mathcal{D}_{\mu})_{l})},\frac{E((\mathcal{D}_{v})_{k})}{ \sum\limits_{l=1}^{s}E((\mathcal{D}_{v})_{l})}\right),\quad k=1,2,\ldots,s,\]  
\[w_{1} =(0.1815,0.1815),w_{2}=(0.1712,0.1712),w_{3}=(0.1592,0.1592),\] \[w_{4} =(0.1642,0.1642),w_{5}=(0.1651,0.1651),w_{6}=(0.1589,0.1589).\]  
Therefore, the weight vector of six experts \(e_{k}\)\((k=1,2,\ldots,6)\) is:  
\(w=((0.1815,0.1815),(0.1712,0.1712),(0.1592,0.1592),(0.1642,0.1642),(0.1651,0.1651 ),(0.1589,0.1589))^{T}\). Figure 6: Pythagorean fuzzy digraphs. Compute the averaged Pythagorean fuzzy element (PFE) \(p_{i}^{(k)}\) of the testing venue \(z_{i}\) over all the other testing venues for the experts \(e_{k}(k=1,2,\ldots,6)\) by the Pythagorean fuzzy averaging (PFA) operator:  
\[p_{i}^{(k)}=PFA(p_{i1}^{(k)},p_{i2}^{(k)},\ldots,p_{in}^{(k)})=\left(\sqrt{1- \left(\prod_{j=1}^{n}\left(1-\mu_{ij}^{2}\right)\right)^{1/n}},\prod_{j=1}^{s} \left(v_{ij}\right)^{1/n}\right),\qquad i=1,2,3,\ldots,n,\]  
The aggregation results are listed in Table 1:  
Compute a collective PFE \(p_{i}\)\((i=1,2,3,4)\) of the testing venue \(z_{i}\) over all the other testing venues using the Pythagorean fuzzy weighted averaging (PFWA) operator [2]. \[p_{i}=\text{PFWA}(p_{i}^{(1)},p_{i}^{(2)},\ldots,p_{i}^{(s)})=\left(\sqrt{1- \prod_{k=1}^{s}\left(1-(\mu_{k}^{2})\right)^{\text{sp}_{k}}},\prod_{k=1}^{s} \left(v_{k}\right)^{v_{k}}\right).\]  
That is, \(p_{1}=(0.6110,0.4752)\), \(p_{2}=(0.5803,0.5203)\), \(p_{3}=(0.5415,0.5550)\), \(p_{4}=(0.6061,0.4814)\).