# Energy of Pythagorean Fuzzy Graphs with Applications
## 4 Energy and Laplacian Energy of Pythagorean Fuzzy Digraphs
: _The Laplacian matrix of a PFDG \(\mathcal{D}=(\mathcal{P},\overrightarrow{\mathcal{Q}})\) is defined as \(L(\mathcal{D})=(L(\mu_{\overrightarrow{\mathcal{Q}}}(z_{i}z_{j})),L(\
u_{ \overrightarrow{\mathcal{Q}}}(z_{i}z_{j})))=D^{out}(\mathcal{D})-A(\mathcal{ D})\), where \(A(\mathcal{D})\) is an adjacency matrix and \(D^{out}(\mathcal{D})\) is an out-degree matrix of a PFDG \(\mathcal{D}\)._  
**Definition 14**. : _The spectrum of the Laplacian matrix of a PFDG \(L(\mathcal{D})\) is defined as \((\mathcal{S}_{L},\mathcal{T}_{L})\), where \(\mathcal{S}_{L}\) and \(\mathcal{T}_{L}\) are the sets of Laplacian eigenvalues of \(L(\mu_{\overrightarrow{\mathcal{Q}}}(z_{i}z_{j}))\) and \(L(\
u_{\overrightarrow{\mathcal{Q}}}(z_{i}z_{j}))\), respectively._  
**Theorem 12**. : _Let \(\mathcal{D}=(\mathcal{P},\overrightarrow{\mathcal{Q}})\) be a PFDG, and let \(L(\mathcal{D})\) be the Laplacian matrix of \(\mathcal{D}\). If \(g_{1}\geq g_{2}\geq\ldots\geq g_{n}\) and \(h_{1}\geq h_{2}\geq\ldots\geq h_{n}\) are the eigenvalues of \(L(\mu_{\overrightarrow{\mathcal{Q}}}(z_{i}z_{j}))\) and \(L(\
u_{\overrightarrow{\mathcal{Q}}}(z_{i}z_{j}))\), respectively, then:_  
\[\sum_{ \begin{subarray}{c}i=1\\ g_{i}\in\mathcal{S}_{L}\end{subarray} }^{n}Re(g_{i})=\sum_{1\leq i<j\leq n}\mu_ {\overrightarrow{\mathcal{Q}}}(z_{i}z_{j}),\sum_{ \begin{subarray}{c}i=1\\ h_{i}\in\mathcal{T}_{L}\end{subarray} }^{n}Re(h_{i})=\sum_{1\leq i<j\leq n}\
u_ {\overrightarrow{\mathcal{Q}}}(z_{i}z_{j}).\]  
**Definition 15**. : _The Laplacian energy of a PFDG \(\mathcal{D}=(\mathcal{P},\overrightarrow{\mathcal{Q}})\) is defined as:_  
\[LE(\mathcal{D})=\left(LE(\mu_{\overrightarrow{\mathcal{Q}}}(z_{i}z_{j})),LE( \
u_{\overrightarrow{\mathcal{Q}}}(z_{i}z_{j}))\right)=\left(\sum_{i=1}^{n}|r_{ i}|_{r}\sum_{i=1}^{n}|s_{i}|\right)\]_where:_  
\[r_{i} = Re(g_{i})-\frac{\sum\limits_{1\leq i<j\leq n}\mu_{\overrightarrow{Q} }(z_{i}z_{j})}{n},\] \[s_{i} = Re(h_{i})-\frac{\sum\limits_{1\leq i<j\leq n}\
u_{\overrightarrow{ Q}}(z_{i}z_{j})}{n}.\]  
**Theorem 13**. : _Let \(\mathcal{D}=(\mathcal{P},\overrightarrow{Q})\) be a PFDG, and let \(L(\mathcal{D})\) be the Laplacian matrix of \(\mathcal{D}\). If \(g_{1}\geq g_{2}\geq\ldots\geq g_{n}\) and \(h_{1}\geq h_{2}\geq\ldots\geq h_{n}\) are the eigenvalues of \(L(\mu_{\overrightarrow{Q}}(z_{i}z_{j}))\) and \(L(\
u_{\overrightarrow{Q}}(z_{i}z_{j}))\), respectively, and \(r_{i}=Re(g_{i})-\frac{\sum\limits_{1\leq i<j\leq n}\
u_{\overrightarrow{Q} }(z_{i}z_{j})}{n},s_{i}=Re(h_{i})-\frac{\sum\limits_{1\leq i<j\leq n}\
u_{ \overrightarrow{Q}}(z_{i}z_{j})}{n}\), then \(\sum\limits_{i=1}^{n}r_{i}=0\), \(\sum\limits_{i=1}^{n}s_{i}=0\)._  
**Example 7**. : _Consider a PFDG \(\mathcal{D}=(\mathcal{P},\overrightarrow{Q})\) on \(V=\{z_{1},z_{2},z_{3},z_{4},z_{5},z_{6}\}\), as shown in Figure 5._  
_The adjacency matrix, out-degree matrix and Laplacian matrix of the PFDG shown in Figure 5 are as follows:_  
\[A(\mathcal{D})=\left[ \begin{array}{cccccc}(0,0)&(0,0)&(0,0)&(0,0)&(0,0)&(0,3,0.6)\\ (0,4,0.7)&(0,0)&(0.3,0.8)&(0,0)&(0,0)&(0,0)\\ (0,0)&(0.5,0.7)&(0,0)&(0,0)&(0.4,0.7)&(0,0)\\ (0,0)&(0,0)&(0.3,0.6)&(0,0)&(0.3,0.8)&(0,0)\\ (0,0)&(0,0)&(0,0)&(0.6,0.5)&(0,0)&(0.7,0.5)\\ (0.6,0.8)&(0.6,0.7)&(0,0)&(0,0)&(0,0)&(0,0)\end{array} \right].\]  
\[D^{out}(\mathcal{D})=\left[ \begin{array}{cccccc}(0.3,0.6)&(0,0)&(0,0)&(0,0)& (0,0)&(-0.3,-0.6)\\ (-0.4,-0.7)&(0.7,1.5)&(-0.3,-0.8)&(0,0)&(0,0)&(0,0)\\ (0,0)&(-0.5,-0.7)&(0.9,1.4)&(0,0)&(-0.4,-0.7)&(0,0)\\ (0,0)&(0,0)&(-0.3,-0.6)&(0.6,1.4)&(-0.3,-0.8)&(0,0)\\ (0,0)&(0,0)&(0,0)&(-0.6,-0.5)&(1.3,1.0)&(-0.7,-0.5)\\ (-0.6,-0.8)&(-0.6,-0.7)&(0,0)&(0,0)&(0,0)&(1.2,1.5)\end{array} \right].\]  
Figure 5: Pythagorean fuzzy digraph.