# Energy of Pythagorean Fuzzy Graphs with Applications
## 4 Energy and Laplacian Energy of Pythagorean Fuzzy Digraphs  
This section generalizes the concept of energy to PFDGs. The eigenvalues of a PFDG may be complex numbers, as its adjacency matrix is not necessarily symmetric. **Definition 10**. : _The spectrum of the adjacency matrix of a PFDG \(A(\mathcal{D})\) is defined as \((\mathcal{S},\mathcal{T})\), where \(\mathcal{S}\) and \(\mathcal{T}\) are the sets of eigenvalues of \(A(\mu_{\overrightarrow{Q}}(z_{i}z_{j}))\) and \(A(\
u_{\overrightarrow{Q}}(z_{i}z_{j}))\), respectively._  
**Definition 11**. : _Let \(\mathcal{D}=(\mathcal{P},\overrightarrow{\mathcal{Q}})\) be a PFDG on \(n\) vertices. The energy of \(\mathcal{D}\) is defined as:_  
\[E(\mathcal{D})=\left(E(\mu_{\overrightarrow{Q}}(z_{i}z_{j})),E(\
u_{ \overrightarrow{Q}}(z_{i}z_{j}))\right)=\left(\sum\limits_{ \begin{subarray}{ c}i=1\\ t_{i}\in\mathcal{S}\end{subarray} }^{n}|Re(t_{i})|,\sum\limits_{ \begin{subarray}{ c}i=1\\ w_{i}\in\mathcal{T}\end{subarray} }^{n}|Re(w_{i})|\right)\]  
_where \(Re(t_{i})\) and \(Re(w_{i})\) represent the real part of eigenvalues \(t_{i}\) and \(w_{i}\), respectively._  
**Theorem 11**. : _Let \(\mathcal{D}=(\mathcal{P},\overrightarrow{\mathcal{Q}})\) be a PFDG and \(A(\mathcal{D})\) be its adjacency matrix. If \(t_{1}\geq t_{2}\geq\ldots\geq t_{n}\) and \(w_{1}\geq w_{2}\geq\ldots\geq w_{n}\) are the eigenvalues of \(A(\mu_{\overrightarrow{Q}}(z_{i}z_{j}))\) and \(A(\
u_{\overrightarrow{Q}}(z_{i}z_{j}))\), respectively, then:_  
\[\sum\limits_{ \begin{subarray}{c}i=1\\ t_{i}\in\mathcal{S}\end{subarray} }^{n}Re(t_{i})=0\;\text{and}\;\;\sum\limits_{ \begin{subarray}{c}i=1\\ w_{i}\in\mathcal{T}\end{subarray} }^{n}Re(w_{i})=0. \tag{10}\]  
**Example 6**. : _Consider a digraph \(D=(V,\overrightarrow{E})\), where \(V=\{z_{1},z_{2},z_{3},z_{4},z_{5},z_{6}\}\) and \(\overrightarrow{E}=\{z_{1}z_{4},z_{1}z_{5},\\ z_{2}z_{4},z_{2}z_{6},z_{3}z_{5},z_{3}z_{6},z_{4}z_{6}\}\). Let \(\mathcal{D}=(\mathcal{P},\overrightarrow{\mathcal{Q}})\) be a PFDG on \(V\), as shown in Figure 4._  
Figure 4: Pythagorean fuzzy digraph. _The adjacency matrix of a PFDG given in Figure 4 is:_  
\[A(\mathcal{D})=\left[ \begin{array}{cccccc}(0,0)&(0,0)&(0,0)&(0.3,0.7)&(0,0)&(0,0 )\\ (0,0)&(0,0)&(0,0)&(0.5,0.8)&(0,0)&(0,0)\\ (0,0)&(0,0)&(0,0)&(0,0)&(0.2,0.7)&(0,0)\\ (0,0)&(0,0)&(0,0)&(0,0)&(0,0)&(0.4,0.6)\\ (0.2,0.9)&(0,0)&(0,0)&(0,0)&(0,0)&(0,0)\\ (0,0)&(0.3,0.9)&(0.6,0.8)&(0,0)&(0,0)&(0,0)\end{array} \right].\]  
_The spectrum and the energy of a PFDG \(\mathcal{D}\) given in Figure 4 are:_  
\[Spec(\mu_{\overrightarrow{\mathcal{Q}}}(z_{i}z_{j}))=\{0.4237,-0.2281+0.3107i,-0.2281-0.3107i,0.0162+0.2133i,0.0162-0.2133i,0\},\] \[Spec(\
u_{\overrightarrow{\mathcal{Q}}}(z_{i}z_{j}))=\{0.8881,-0. 5478+0.5900i,-0.5478-0.5900i,0.1037+0.5975i,0.1037-0.5975i,0\}.\]  
_Therefore,_  
\[Spec(\mathcal{D}) = \{(0.4237,0.8881),(-0.2281+0.3107i,-0.5478+0.5900i),(-0.2281-0.31 07i,-0.5478-0.5900i),\] \[(0.0162+0.2133i,0.1037+0.5975i),(0.0162-0.2133i,0.1037-0.5975i),(0,0)\}.\]  
_Now, \(E(\mu_{\overrightarrow{\mathcal{Q}}}(z_{i}z_{j}))=0.9123\) and \(E(\
u_{\overrightarrow{\mathcal{Q}}}(z_{i}z_{j}))=2.1910\)._  
_Therefore,_  
\[\sum_{ \begin{subarray}{c}i=1\\ i_{1}\in\mathcal{S}\end{subarray} }^{6}Re(t_{i})=0\;\text{and}\;\;\sum_{ \begin{subarray}{c}i=1\\ w_{i}\in\mathcal{T}\end{subarray} }^{6}Re(w_{i})=0. \tag{11}\]  
We now discuss the Laplacian energy of Pythagorean fuzzy digraphs. **Definition 12**. : _Let \(\mathcal{D}=(\mathcal{P},\overrightarrow{\mathcal{Q}})\) be a PFDG on \(n\) vertices. The degree matrix, \(D(\mathcal{D})=(D(\mu_{\overrightarrow{\mathcal{Q}}}(z_{i}z_{j})),D(\
u_{ \overrightarrow{\mathcal{Q}}}(z_{i}z_{j})))=[d_{ij}]\), of \(\mathcal{D}\) is a \(n\times n\) diagonal matrix defined as:_  
\[d_{ij}=\left\{ \begin{array}{ll}d_{\mathcal{D}}^{out}(z_{i})&\text{if}\;i= j,\\ 0&\text{otherwise}\end{array} \right.\]  
**Definition 13**.