## 3 Edge Regularity of a Graph in Complex Pythagorean Fuzzy Circumstances
}\
u_{\mathcal{D}}(r_{i}r_{k})e^{ \begin{subarray}{c}i \sum\\ k\
eq j\end{subarray} }+\sum_{ \begin{subarray}{c}r_{j}r_{k}\in D\\ k\
eq i\end{subarray} }\
u_{\mathcal{D}}(r_{j}r_{k})e^{ \begin{subarray}{c}i \sum\\ k\
eq i\end{subarray} }+\
u_{\mathcal{D}}(r_{i}r_{j})e^{i\beta_{\mathcal{D}}(r_{i}r_{j })}=\mathcal{S}_{2}.\]  
\(\mathcal{G}\) _is called totally edge regular of degree \((\mathcal{S}_{1},\mathcal{S}_{2})\) or \((\mathcal{S}_{1},\mathcal{S}_{2})\)-totally edge regular._  
**Example 9**. : _Consider a CPFG \(\mathcal{G}=(\mathcal{C},\mathcal{D})\) on \(C=\{s_{1},s_{2},s_{3},s_{4}\}\), as in Figure 10, defined by:_  
\[\mathcal{C} = \left\langle\left(\frac{s_{1}}{0.8e^{2\pi(0.7)}},\frac{s_{2}}{0. 7e^{2\pi(0.8)}},\frac{s_{3}}{0.6e^{2\pi(0.9)}},\frac{s_{4}}{0.6e^{2\pi(0.7)}} \right),\left(\frac{s_{1}}{0.5e^{2\pi(0.6)}},\frac{s_{2}}{0.2e^{2\pi(0.3)}}, \frac{s_{3}}{0.4e^{2\pi(0.5)}},\frac{s_{4}}{0.3e^{2\pi(0.5)}}\right)\right\rangle,\] \[\mathcal{D} = \left\langle\left(\frac{s_{1}s_{2}}{0.4e^{2\pi(0.6)}},\frac{s_{2}s_ {3}}{0.4e^{2\pi(0.6)}},\frac{s_{3}s_{4}}{0.4e^{2\pi(0.6)}},\frac{s_{4}s_{1}}{0.4 e^{2\pi(0.6)}}\right),\left(\frac{s_{1}s_{2}}{0.1e^{2\pi(0.5)}},\frac{s_{2}s_{3}}{0.1e^{2 \pi(0.5)}},\frac{s_{3}s_{4}}{0.1e^{2\pi(0.5)}},\frac{s_{4}s_{1}}{0.1e^{2\pi(0.5)}} \right)\right\rangle.\]_Clearly, \(\mathsf{td}_{\mathcal{G}}(s_{1}s_{2})=\mathsf{td}_{\mathcal{G}}(s_{2}s_{3})= \mathsf{td}_{\mathcal{G}}(s_{3}s_{4})=\mathsf{td}_{\mathcal{G}}(s_{4}s_{1})=(1.2e ^{i2\pi(1.8)},0.3e^{i2\pi(1.5)})\). So, \(\mathcal{G}\) is \((1.2e^{i2\pi(1.8)},0.3e^{i2\pi(1.5)})\)-totally regular CPFG._  
**Theorem 10**. : _Let \(\mathcal{G}=(\mathcal{C},\mathcal{D})\) be a CPFG on a regular graph \(G\). Then_  
\[\sum_{r:r_{j}\in D}\mathsf{td}_{\mathcal{G}}(r_{i}r_{j}) = \left(\sum_{r:r_{j}\in D}\mathsf{d}_{\mathcal{G}}(r_{i}r_{j})\mu _{\mathcal{D}}(r_{i}r_{j})e^{i\alpha_{\mathcal{G}}(r_{i}r_{j})}+\sum_{r:r_{j} \in D}\mu_{\mathcal{G}}(r_{i}r_{j})e^{i\sum\limits_{r^{\prime}_{i}r^{\prime}_{ j}\in D}\alpha_{\mathcal{G}}(r_{i}r_{j})},\right.\] \[\left.\sum_{r_{i}r_{j}\in D}\mathsf{d}_{\mathcal{G}}(r_{i}r_{j}) \
u_{\mathcal{G}}(r_{i}r_{j})e^{i\beta_{\mathcal{G}}(r_{i}r_{j})}+\sum_{r:r_{ j}\in D}\
u_{\mathcal{G}}(r_{i}r_{j})e^{i\sum\limits_{r^{\prime}_{i}r^{\prime}_{ j}\in D}\beta_{\mathcal{G}}(r_{i}r_{j})}\right).\]  
**Theorem 11**. : _Let \(\mathcal{G}=(\mathcal{C},\mathcal{D})\) be a CPFG. Then \((\mu_{\mathcal{D}}e^{i\alpha_{\mathcal{D}}},\
u_{\mathcal{D}}e^{i\beta_{ \mathcal{D}}})\) is a constant function if and only if the statements given below are equivalent:_  
**(i)**: \(\mathcal{G}\) _is an edge regular CPFG;_
**(ii)**: \(\mathcal{G}\) _is a totally edge regular CPFG._  
Proof. : Assume that \((\mu_{\mathcal{D}}e^{i\alpha_{\mathcal{D}}},\
u_{\mathcal{D}}e^{i\beta_{ \mathcal{D}}})\) is a constant function. Then \(\mu_{\mathcal{D}}(r_{i}r_{j})e^{i\alpha_{\mathcal{G}}(r_{i}r_{j})}=c_{1}e^{i2 \pi(c^{\prime}_{1})}\) and \(\
u_{\mathcal{G}}(r_{i}r_{j})e^{i\beta_{\mathcal{G}}(r_{i}r_{j})}=c_{2}e^{i2 \pi(c^{\prime}_{2})}\) for every \(r_{i}r_{j}\in D\), where \(c_{1}e^{i2\pi(c^{\prime}_{1})}\) and \(c_{2}e^{i2\pi(c^{\prime}_{2})}\) are constants. \((i)\Rightarrow(ii)\). Suppose that \(\mathcal{G}\) is \((s_{1},s_{2})\)-edge regular CPFG. Then \(\mathsf{d}_{\mathcal{G}}(r_{i}r_{j})=(s_{1},s_{2})\) for all \(r_{i}r_{j}\in D\). Consider  
\[\left(\mathsf{d}_{\mu e^{i\alpha}}(r_{i}r_{j})+\mu_{\mathcal{D}}(r _{i}r_{j})e^{i\alpha_{\mathcal{G}}(r_{i}r_{j})},\mathsf{d}_{\
u e^{i\beta}}(r _{i}r_{j})+\
u_{\mathcal{D}}(r_{i}r_{j})e^{i\beta_{\mathcal{G}}(r_{i}r_{j})}\right)\] \[=(s_{1}+c_{1}e^{i2\pi(c^{\prime}_{1})},s_{2}+c_{2}e^{i2\pi(c^{ \prime}_{2})})\text{ for all }r_{i}r_{j}\in D.\]  
Therefore, CPFG \(\mathcal{G}\) is a totally edge regular. \((ii)\Rightarrow(i)\). Let \(\mathcal{G}\) be a \((\mathcal{S}_{1},\mathcal{S}_{2})\)-totally edge regular CPFG.