# Energy of Pythagorean Fuzzy Graphs with Applications
## 2 Energy of Pythagorean Fuzzy Graphs  
In this section, we define the energy of a graph under Pythagorean fuzzy circumstances and investigate its properties. Definition 2 ([27]): A PFG on a non-empty set \(Z\) is a pair \(\mathcal{G}=(\mathcal{P},\mathcal{Q})\), where \(\mathcal{P}\) is a PFS on \(Z\) and \(\mathcal{Q}\) is a Pythagorean fuzzy relation on \(Z\) such that:  
\[\mu_{\mathcal{Q}}(yz)\leq\min\{\mu_{\mathcal{P}}(y),\mu_{\mathcal{P}}(z)\}, \
u_{\mathcal{Q}}(yz)\geq\max\{\
u_{\mathcal{P}}(y),\
u_{\mathcal{P}}(z)\}\]  
and \(0\leq\mu_{\mathcal{Q}}^{2}(yz)+\
u_{\mathcal{Q}}^{2}(yz)\leq 1\) for all \(y,z\in Z\). We call \(\mathcal{P}\) and \(\mathcal{Q}\) the Pythagorean fuzzy vertex set and the Pythagorean fuzzy edge set of \(\mathcal{G}\), respectively. Here, \(\mathcal{Q}\) is a symmetric Pythagorean fuzzy relation on \(\mathcal{P}\). If \(\mathcal{Q}\) is not symmetric on \(\mathcal{P}\), then \(\mathcal{D}=(\mathcal{P},\overrightarrow{Q})\) is called PFDG. Example 1: Consider a graph \(G=(V,E)\), where \(V=\{z_{1},z_{2},z_{3},z_{4},z_{5},z_{6}\}\) is the vertex set and \(E=\{z_{1}z_{2},z_{1}z_{3},z_{1}z_{4},z_{1}z_{5},z_{1}z_{6},z_{2}z_{3},z_{3}z_{4 },z_{4}z_{5},z_{5}z_{6}\}\) is the edge set of \(G\). Let \(\mathcal{G}=(\mathcal{P},\mathcal{Q})\) be a PFG on \(V\), as shown in Figure 2, defined by:  
\[\mathcal{P} = \left\langle\left(\frac{z_{1}}{0.8},\frac{z_{2}}{0.7},\frac{z_{3} }{0.5},\frac{z_{4}}{0.8},\frac{z_{5}}{0.7},\frac{z_{6}}{0.4}\right),\left( \frac{z_{1}}{0.5},\frac{z_{2}}{0.4},\frac{z_{3}}{0.6},\frac{z_{4}}{0.3},\frac{ z_{5}}{0.4},\frac{z_{6}}{0.3}\right)\right\rangle,\] \[\mathcal{Q} = \left\langle\left(\frac{z_{1}z_{2}}{0.3},\frac{z_{1}z_{3}}{0.4},\frac{z_{1}z_{4}}{0.7},\frac{z_{1}z_{5}}{0.4},\frac{z_{1}z_{6}}{0.3},\frac{ z_{2}z_{3}}{0.4},\frac{z_{3}z_{4}}{0.5},\frac{z_{4}z_{5}}{0.6},\frac{z_{5}z_{6}}{0.4} \right),\right.\] \[\left.\left(\frac{z_{1}z_{2}}{0.8},\frac{z_{1}z_{3}}{0.7},\frac{z_ {1}z_{4}}{0.6},\frac{z_{1}z_{5}}{0.5},\frac{z_{1}z_{6}}{0.9},\frac{z_{2}z_{3}}{ 0.7},\frac{z_{3}z_{4}}{0.6},\frac{z_{5}z_{6}}{0.9}\right)\right\rangle.\]  
Figure 1: Comparison of spaces of the IFNs and the Pythagorean fuzzy numbers (PFNs). Figure 2: Pythagorean fuzzy graph. **Definition 3**. : _The adjacency matrix \(A(\mathcal{G})=(A(\mu_{\mathcal{Q}}(z_{i}z_{j})),A(\
u_{\mathcal{Q}}(z_{i}z_{j})))\) of a PFG \(\mathcal{G}=(\mathcal{P},\mathcal{Q})\) is defined as a square matrix \(A(\mathcal{G})=[a_{ij}],a_{ij}=(\mu_{\mathcal{Q}}(z_{i}z_{j}),\
u_{\mathcal{Q}} (z_{i}z_{j}))\), where \(\mu_{\mathcal{Q}}(z_{i}z_{j})\) and \(\
u_{\mathcal{Q}}(z_{i}z_{j})\) represent the strength of relationship and strength of non-relationship between \(z_{i}\) and \(z_{j}\), respectively._  
**Definition 4**. : _The spectrum of adjacency matrix of a PFG \(A(\mathcal{G})\) is defined as \((S,T)\), where \(S\) and \(T\) are the sets of eigenvalues of \(A(\mu_{\mathcal{Q}}(z_{i}z_{j}))\) and \(A(\
u_{\mathcal{Q}}(z_{i}z_{j}))\), respectively._  
**Definition 5**. : _The energy of a PFG \(\mathcal{G}=(\mathcal{P},\mathcal{Q})\) is defined as:_  
\[E(\mathcal{G})=\left(E(\mu_{\mathcal{Q}}(z_{i}z_{j})),E(\
u_{\mathcal{Q}}(z_{ i}z_{j}))\right)=\left(\sum_{ \begin{subarray}{c}i=1\\ \lambda_{i}\in S\end{subarray} }^{n}|\lambda_{i}|,\sum_{ \begin{subarray}{c}i=1 \\ \zeta_{i}\in T\end{subarray} }^{n}|\zeta_{i}|\right).\]  
**Example 2**. : _The adjacency matrix of a PFG given in Figure 2 is:_  
\[A(\mathcal{G})=\left[ \begin{array}{cccccc}(0,0)&(0.3,0.8)&(0.4,0.7)&(0.7,0.6 )&(0.4,0.5)&(0.3,0.6)\\ (0.3,0.8)&(0,0)&(0.4,0.9)&(0,0)&(0,0)&(0,0)\\ (0.4,0.7)&(0.4,0.9)&(0,0)&(0.5,0.7)&(0,0)&(0,0)\\ (0.7,0.6)&(0,0)&(0.5,0.7)&(0,0)&(0.6,0.6)&(0,0)\\ (0.4,0.5)&(0,0)&(0,0)&(0.6,0.6)&(0,0)&(0.4,0.9)\\ (0.3,0.6)&(0,0)&(0,0)&(0.4,0.9)&(0,0)\end{array}