# Energy of Pythagorean Fuzzy Graphs with Applications
## 3 Laplacian Energy of Pythagorean Fuzzy Graphs  
This section defines and investigates the Laplacian energy of a PFG and provides its properties in detail. **Definition 6**. : _Let \(\mathcal{G}=(\mathcal{P},\mathcal{Q})\) be a PFG on \(n\) vertices. The degree matrix, \(D(\mathcal{G})=(D(\mu_{Q}(z_{i}z_{j})),D(\
u_{Q}(z_{i}z_{j})))=[d_{ij}]\), of \(\mathcal{G}\) is a \(n\times n\) diagonal matrix defined as:_  
\[d_{ij}=\left\{ \begin{array}{ll}d_{\mathcal{G}}(z_{i})&\text{ if }i=j,\\ 0&\text{ otherwise}\end{array} \right.\]  
**Definition 7**. : _The Laplacian matrix of a PFG \(\mathcal{G}=(\mathcal{P},\mathcal{Q})\) is defined as \(L(\mathcal{G})=(L(\mu_{Q}(z_{i}z_{j})),L(\
u_{Q}(z_{i}z_{j})))=D(\mathcal{G} )-A(\mathcal{G})\), where \(A(\mathcal{G})\) is an adjacency matrix and \(D(\mathcal{G})\) is a degree matrix of a PFG \(\mathcal{G}\)._  
**Example 4**. : _Consider a graph \(G=(V,E)\), where \(V=\{z_{1},z_{2},z_{3},z_{4},z_{5},z_{6},z_{7},z_{8}\}\) and \(E=\{z_{1}z_{2},z_{2}z_{3},\)\(z_{3}z_{4},z_{1}z_{4},z_{2}z_{4},z_{2}z_{8},z_{4}z_{6},z_{1}z_{8},z_{4}z_{5},z_{5 }z_{6},z_{6}z_{7},z_{7}z_{8},z_{5}z_{8},z_{6}z_{8}\}\). Let \(\mathcal{G}=(\mathcal{P},\mathcal{Q})\) be a PFG on \(V\), as shown in Figure 3._  
_The adjacency matrix, degree matrix and Laplacian matrix of the PFG shown in Figure 3 are as follows:_  
\[A(\mathcal{G})=\left[ \begin{array}{cccccccc}(0,0)&(0.4,0.7)&(0,0)&(0.6,0.7)&( 0,0)&(0,0)&(0,0)&(0.5,0.7)\\ (0.4,0.7)&(0,0)&(0.4,0.6)&(0.5,0.6)&(0,0)&(0,0)&(0,0)&(0.3,0.7)\\ (0,0)&(0.4,0.6)&(0,0)&(0.6,0.8)&(0,0)&(0,0)&(0,0)&(0,0)\\ (0.6,0.7)&(0.5,0.6)&(0.6,0.8)&(0,0)&(0.4,0.7)&(0.6,0.8)&(0,0)&(0,0)\\ (0,0)&(0,0)&(0,0)&(0.4,0.7)&(0,0)&(0.4,0.9)&(0,0)&(0.3,0.6)\\ (0,0)&(0,0)&(0,0)&(0.6,0.8)&(0.4,0.9)&(0,0)&(0.4,0.8)&(0.3,0.9)\\ (0,0)&(0,0)&(0,0)&(0,0)&(0.4,0.8)&(0,0)&(0.2,0.6)\\ (0.5,0.7)&(0.3,0.7)&(0,0)&(0,0)&(0.3,0.6)&(0.3,0.9)&(0.2,0.6)&(0,0)\\ \end{array} \right].\]  
\[D(\mathcal{G})=\left[ \begin{array}{cccccccc}(1.5,2.1)&(0,0)&(0,0)&(0,0)&(0,0) &(0,0)&(0,0)&(0,0)\\ (0,0)&(1.6,2.6)&(0,0)&(0,0)&(0,0)&(0,0)&(0,0)&(0,0)\\ (0,0)&(0,0)&(1.0,1.4)&(0,0)&(0,0)&(0,0)&(0,0)&(0,0)\\ (0,0)&(0,0)&(0,0)&(2.7,3.6)&(0,0)&(0,0)&(0,0)&(0,0)\\ (0,0)&(0,0)&(0,0)&(1.1,2.2)&(0,0)&(0,0)&(0,0)\\ (0,0)&(0,0)&(0,0)&(0,0)&(1.7,3.4)&(0,0)&(0,0)\\ (0,0)&(0,0)&(0,0)&(0,0)&(0,0)&(0,6.1.4)&(0,0)\\ (0,0)&(0,0)&(0,0)&(0,0)&(0,0)&(0,0)&(1.6,3.5)\\ \end{array} \right].\]  
\[L(\mathcal{G})=\left[ \begin{array}{cccccccc}(1.5,2.1)&(-0.4,-0.7)&(0,0)&(-0. 6,-0.7)&(0,0)&(0,0)&(0,0)&(-0.5,-0.7)\\ (-0.4,-0.7)&(1.6,2.6)&(-0.4,-0.6)&(-0.5,-0.6)&(0,0)&(0,0)&(-0.3,-0.7)\\ (0,0)&(-0.4,-0.6)&(1.0,1.4)&(-0.6,-0.8)&(0,0)&(0,0)&(0,0)\\ (-0.6,-0.7)&(-0.5,-0.6)&(-0.6,-0.8)&(2.7,3.6)&(-0.4,-0.7)&(-0.6,-0.8)&(0,0)&(0,0)\\ (0,0)&(0,0)&(0,0)&(-0.4,-0.7)&(1.1,2.2)&(-0.4,-0.9)&(0,0)&(-0.3,-0.6)\\ (0,0)&(0,0)&(0,0)&(-0.6,-0.8)&(-0.4,-0.9)&(1.7,3.4)&(-0.4,-0.8)&(-0.3,-0.9)\\ (0,0)&(0,0)&(0,0)&(0,0)&(-0.4,-0.8)&(0.6,1.4)&(-0.2,-0.6)\\ (-0.5,-0.7)&(-0.3,-0.7)&(0,0)&(0,0)&(-0.3,-0.6)&(-0.3,-0.9)&(-0.2,-0.6)&(1.6,3.5)\\ \end{array} \right].\]  
**Definition 8**. : _The spectrum of Laplacian matrix of a PFG \(L(\mathcal{G})\) is defined as \((S_{L},T_{L})\), where \(S_{L}\) and \(T_{L}\) are the sets of Laplacian eigenvalues of \(L(\mu_{Q}(z_{i}z_{j}))\) and \(L(\
u_{Q}(z_{i}z_{j}))\), respectively._  
Figure 3. Pythagorean fuzzy graph. **Theorem 5**. : _Let \(\mathcal{G}=(\mathcal{P},\mathcal{Q})\) be a PFG, and let \(L(\mathcal{G})\) be the Laplacian matrix of \(\mathcal{G}\). If \(\vartheta_{1}\geq\vartheta_{2}\geq\ldots\geq\vartheta_{n}\) and \(\varphi_{1}\geq\varphi_{2}\geq\ldots\geq\varphi_{n}\) are the eigenvalues of \(L(\mu_{Q}(z_{i}z_{j}))\) and \(L(\
u_{Q}(z_{i}z_{j}))\), respectively, then:_  
**(i)**: \(\sum\limits_{ \begin{subarray}{c}i=1\\ \vartheta_{i}\in S_{L}\end{subarray} }^{n}\vartheta_{i}=2\sum\limits_{1\leq i< j\leq n}\mu_{Q}(z_{i}z_{j})\)_,_ \(\sum\limits_{ \begin{subarray}{c}i=1\\ \vartheta_{i}\in T_{L}\end{subarray} }^{n}\varphi_{i}=2\sum\limits_{1\leq i< j\leq n}\
u_{Q}(z_{i}z_{j})\)_._
**(ii)**: \(\sum\limits_{