# Energy of Pythagorean Fuzzy Graphs with Applications
## 2 Energy of Pythagorean Fuzzy Graphs
\right].\]  
_The spectrum and the energy of a PFG \(\mathcal{G}\), given in Figure 2, are as follows:_  
\[Spec(\mu_{\mathcal{Q}}(z_{i}z_{j}))=\{-0.9522,-0.5665,-0.3997,0.0031,0.4017,1.5137\},\] \[Spec(\
u_{\mathcal{Q}}(z_{i}z_{j}))=\{-1.3553,-0.9082,-0.8959,0. 0534,0.9017,2.2044\}.\]  
_Therefore,_  
\[Spec(\mathcal{G}) = \{(-0.9522,-1.3553),(-0.5665,-0.9082),(-0.3997,-0.8959),(0.0031,0.0534),\] \[(0.4017,0.9017),(1.5137,2.2044)\}.\]  
_Now, \(E(\mu_{\mathcal{Q}}(z_{i}z_{j}))=3.8369\) and \(E(\
u_{\mathcal{Q}}(z_{i}z_{j}))=6.3190\)._  
_Therefore, \(E(\mathcal{G})=(3.8369,6.3190)\)._  
**Theorem 1**. : _Let \(\mathcal{G}=(\mathcal{P},\mathcal{Q})\) be a PFG and \(A(\mathcal{G})\) be its adjacency matrix. If \(\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{n}\) and \(\zeta_{1}\geq\zeta_{2}\geq\ldots\geq\zeta_{n}\) are the eigenvalues of \(A(\mu_{\mathcal{Q}}(z_{i}z_{j}))\) and \(A(\
u_{\mathcal{Q}}(z_{i}z_{j}))\), respectively, then:_  
**(i)**: \(\sum\limits_{ \begin{subarray}{c}i=1\\ \lambda_{i}\in S\end{subarray} }^{n}\lambda_{i}=0\) _and_ \(\sum\limits_{ \begin{subarray}{c}i=1\\ \zeta_{i}\in T\end{subarray} }^{n}\zeta_{i}=0\)_._
**(ii)**: \(\sum\limits_{ \begin{subarray}{c}i=1\\ \lambda_{i}\in S\end{subarray} }^{n}\lambda_{i}^{2}=2\sum\limits_{1\leq i<j\leq n }(\mu_{\mathcal{Q}}(z_{i}z_{j}))^{2}\) _and_ \(\sum\limits_{ \begin{subarray}{c}i=1\\ \zeta_{i}\in T\end{subarray} }^{n}\zeta_{i}^{2}=2\sum\limits_{1\leq i<j\leq n} (\
u_{\mathcal{Q}}(z_{i}z_{j}))^{2}\)_._  
Proof. :
**(i)**: Since \(A(\mathcal{G})\) is a symmetric matrix with zero trace, its eigenvalues are real with the sum equal to zero. **(ii)**: By the trace properties of a matrix, we have:  
\[tr((A(\mu_{\mathcal{Q}}(z_{i}z_{j})))^{2})=\sum\limits_{ \begin{subarray}{c}i=1 \\ \lambda_{i}\in S\end{subarray} }^{n}\lambda_{i}^{2}\]where:  
\[tr((A(\mu_{\mathcal{Q}}(z_{i}z_{j})))^{2}) = \Big{(}0+(\mu_{\mathcal{Q}}(z_{1}z_{2}))^{2}+\ldots+(\mu_{\mathcal{ Q}}(z_{1}z_{n}))^{2}\Big{)}\] \[+\left((\mu_{\mathcal{Q}}(z_{2}z_{1}))^{2}+0+\ldots+(\mu_{ \mathcal{Q}}(z_{2}z_{n}))^{2}\right)\] \[\vdots\] \[+\left((\mu_{\mathcal{Q}}(z_{n}z_{1}))^{2}+(\mu_{\mathcal{Q}}(z_ {n}z_{2}))^{2}+\ldots+0\right)\] \[= 2\sum_{1\leq i<j\leq n}(\mu_{\mathcal{Q}}(z_{i}z_{j}))^{2}.\]  
Hence:  
\[\sum_{ \begin{subarray}{c}i=1\\ \lambda_{i}\in S\end{subarray} }^{n}\lambda_{i}^{2}=2\sum_{1\leq i<j\leq n}(\mu _{\mathcal{Q}}(z_{i}z_{j}))^{2}.\]  
Analogously, we can show that \(\sum\limits_{ \begin{subarray}{c}j=1\\ \xi_{i}\in T\end{subarray} }^{n}\zeta_{i}^{2}=2\sum\limits_{1\leq i<j\leq n}( \
u_{\mathcal{Q}}(z_{i}z_{j}))^{2}\). **Example 3**. : _Consider a PFG \(\mathcal{G}=(\mathcal{P},\mathcal{Q})\) on \(V=\{z_{1},z_{2},z_{3},z_{4},z_{5},z_{6}\}\), as shown in Figure 2. Then:_  
\[\sum_{ \begin{subarray}{c}i=1\\ \lambda_{i}\in S\end{subarray} }^{6}\lambda_{i}=0,\ \sum_{ \begin{subarray}{c}i=1\\ \xi_{i}\in T\end{subarray} }^{6}\zeta_{i}=0.\] \[\sum_{ \begin{subarray}{c}i=1\\ \lambda_{i}\in S\end{subarray} }^{6}\lambda_{i}^{2}=3.8400=2(1.92)=2\sum_{1\leq i <j\leq 6}(\mu_{\mathcal{Q}}(z_{i}z_{j}))^{2},\] \[\sum_{ \begin{subarray}{c}i=1\\ \xi_{i}\in T\end{subarray} }^{6}\zeta_{i}^{2}=9.1400=2(4.57)=2\sum_{1\leq i<j \leq 6}(\
u_{\mathcal{Q}}(z_{i}z_{j}))^{2}.\]  
We now find upper and lower bounds of the energy of a PFG \(\mathcal{G}\), in terms of the number of vertices and the sum of squares of membership and non-membership values of edges. **Theorem 2**. : _Let \(\mathcal{G}=(\mathcal{P},\mathcal{Q})\) be a PFG on \(n\) vertices and \(A(\mathcal{G})=(A(\mu_{\mathcal{Q}}(z_{i}z_{j})),A(\
u_{\mathcal{Q}}(z_{i}z_{ j})))\) be the adjacency matrix of \(\mathcal{G}\).