# Energy of Pythagorean Fuzzy Graphs with Applications
## 3 Laplacian Energy of Pythagorean Fuzzy Graphs
\begin{subarray}{c}i=1\\ \vartheta_{i}\in S_{L}\end{subarray} }^{n}\vartheta_{i}^{2}=2\sum\limits_{1\leq i <j\leq n}(\mu_{Q}(z_{i}z_{j}))^{2}+\sum\limits_{i=1}^{n}d_{\mu_{Q}(z_{i}z_{j} )}^{2}(z_{i})\)_,_ \(\sum\limits_{ \begin{subarray}{c}i=1\\ \vartheta_{i}\in T_{L}\end{subarray} }^{n}\varphi_{i}^{2}=2\sum\limits_{1\leq i <j\leq n}(\
u_{Q}(z_{i}z_{j}))^{2}+\sum\limits_{i=1}^{n}d_{\
u_{Q}(z_{i}z_{j} )}^{2}(z_{i})\)_._  
Proof. : **(i)**: Since \(L(\mathcal{G})\) is a symmetric matrix with non-negative Laplacian eigenvalues, such that:  
\[\sum\limits_{ \begin{subarray}{c}i=1\\ \vartheta_{i}\in S_{L}\end{subarray} }^{n}\vartheta_{i}=tr(L(\mathcal{G}))= \sum\limits_{i=1}^{n}d_{\mu_{Q}(z_{i}z_{j})}(z_{i})=2\sum\limits_{1\leq i<j \leq n}\mu_{Q}(z_{i}z_{j}).\]  
Therefore, \(\sum\limits_{ \begin{subarray}{c}i=1\\ \vartheta_{i}\in S_{L}\end{subarray} }^{n}\vartheta_{i}=2\sum\limits_{1\leq i <j\leq n}\mu_{Q}(z_{i}z_{j})\). Similarly, it is easy to show that, \(\sum\limits_{ \begin{subarray}{c}i=1\\ \vartheta_{i}\in T_{L}\end{subarray} }^{n}\varphi_{i}=2\sum\limits_{1\leq i<j \leq n}\
u_{Q}(z_{i}z_{j})\). **(ii)**: By the definition of Laplacian matrix, we have:  
\[L(\mu_{Q}(z_{i}z_{j}))=\left[ \begin{array}{cccc}d_{\mu_{Q}(z_{i}z_{j})}(z_{ 1})&-\mu_{Q}(z_{1}z_{2})&\ldots&-\mu_{Q}(z_{1}z_{n})\\ -\mu_{Q}(z_{2}z_{1})&d_{\mu_{Q}(z_{i}z_{j})}(z_{2})&\ldots&-\mu_{Q}(z_{2}z_{n} )\\ \vdots&\vdots&\ddots&\vdots\\ -\mu_{Q}(z_{n}z_{1})&-\mu_{Q}(z_{n}z_{2})&\ldots&d_{\mu_{Q}(z_{i}z_{j})}(z_{n} )\end{array} \right].\]  
By the trace properties of a matrix, we have:  
\[tr((L(\mu_{Q}(z_{i}z_{j})))^{2})=\sum\limits_{ \begin{subarray}{c}i=1\\ \vartheta_{i}\in S_{L}\end{subarray} }^{n}\vartheta_{i}^{2}\]  
where:  
\[tr((L(\mu_{Q}(z_{i}z_{j})))^{2}) = (d_{\mu_{Q}(z_{i}z_{j})}^{2}(z_{1})+\mu_{Q}^{2}(z_{1}z_{2})+ \ldots+\mu_{Q}^{2}(z_{1}z_{n}))\] \[+(\mu_{Q}^{2}(z_{2}z_{1})+d_{\mu_{Q}(z_{i}z_{j})}^{2}(z_{2})+ \ldots+\mu_{Q}^{2}(z_{2}z_{n}))\] \[+\ldots+(\mu_{Q}^{2}(z_{n}z_{1})+\mu_{Q}^{2}(z_{n}z_{2})+\ldots+d _{\mu_{Q}(z_{i}z_{j})}^{2}(z_{n}))\] \[= 2\sum\limits_{1\leq i<j\leq n}(\mu_{Q}(z_{i}z_{j}))^{2}+\sum \limits_{i=1}^{n}d_{\mu_{Q}(z_{i}z_{j})}^{2}(z_{i}).\]  
Therefore,  
\[\sum\limits_{ \begin{subarray}{c}i=1\\ \vartheta_{i}\in S_{L}\end{subarray} }^{n}\vartheta_{i}^{2}=2\sum\limits_{1\leq i <j\leq n}(\mu_{Q}(z_{i}z_{j}))^{2}+\sum\limits_{i=1}^{n}d_{\mu_{Q}(z_{i}z_{j })}^{2}(z_{i}).\]  
Analogously, we can show that \(\sum\limits_{ \begin{subarray}{c}i=1\\ \vartheta_{i}\in T_{L}\end{subarray} }^{n}\varphi_{i}^{2}=2\sum\limits_{1\leq i <j\leq n}(\
u_{Q}(z_{i}z_{j}))^{2}+\sum\limits_{i=1}^{n}d_{\
u_{Q}(z_{i}z_{j} )}^{2}(z_{i})\). **Definition 9**. : _The Laplacian energy of a PFG \(\mathcal{G}=(\mathcal{P},\mathcal{Q})\) is defined as:_  
\[LE(\mathcal{G})=\left(LE(\mu_{\mathcal{Q}}(z_{i}z_{j})),LE(\
u_{\mathcal{Q}}(z_{ i}z_{j}))\right)=\left(\sum\limits_{i=1}^{n}|\varrho_{i}|,\sum\limits_{i=1}^{n}| \xi_{i}|\right)\]  
_where:_  
\[\varrho_{i} = \vartheta_{i}-\frac{2\sum\limits_{1\leq i<j\leq n}\mu_{\mathcal{Q }}(z_{i}z_{j})}{n},\] \[\xi_{i} = \varphi_{i}-\frac{2\sum\limits_{1\leq i<j\leq n}\
u_{\mathcal{Q}} (z_{i}z_{j})}{n}.\]  
**Theorem 6**. : _Let \(\mathcal{G}=(\mathcal{P},\mathcal{Q})\) be a PFG, and let \(L(\mathcal{G})\) be the Laplacian matrix of \(\mathcal{G}\).