## 3 Edge Regularity of a Graph in Complex Pythagorean Fuzzy Circumstances
\begin{subarray}{c}i \sum\limits_{r_{i}r_{k}\in D}\alpha_{\mathcal{G}}(r_{j}r_{k})\\ k\
eq i\end{subarray} }\] \[= \sum_{ \begin{subarray}{c}r_{i}r_{k}\in D\\ k\
eq j\end{subarray} }c_{1}e^{ \begin{subarray}{c}2\pi(\sum\limits_{r_{i}r_{k} \in D}c_{1}^{\prime})\\ k\
eq j\end{subarray} }+\sum_{ \begin{subarray}{c}r_{i}r_{k}\in D\\ k\
eq i\end{subarray} }c_{1}e^{ \begin{subarray}{c}i2\pi(\sum\limits_{r_{i}r_{k} \in D}c_{1}^{\prime})\\ k\
eq i\end{subarray} }\] \[= c_{1}e^{2\pi(c_{1}^{\prime})}(\mathrm{d}_{G}(r_{i})-1)+c_{1}e^{ 2\pi(c_{1}^{\prime})}(\mathrm{d}_{G}(r_{j})-1)\] \[= c_{1}e^{i2\pi(c_{1}^{\prime})}(\mathrm{d}_{G}(r_{i})+\mathrm{d}_{ G}(r_{j})-2)\] \[= c_{1}e^{i2\pi(c_{1}^{\prime})}\mathrm{d}_{G}(r_{i}r_{j})\text{ for all }r_{i}r_{j}\in D.\]  
Analogously, we can show that \(\mathrm{d}_{vv^{i\beta}}(r_{i}r_{j})=c_{2}e^{i2\pi(c_{2}^{\prime})}\mathrm{d} _{G}(r_{i}r_{j})\) for all \(r_{i}r_{j}\in D\). Therefore \(\mathrm{d}_{\mathcal{G}}(r_{i}r_{j})=(c_{1}e^{i2\pi(c_{1}^{\prime})}\mathrm{d }_{G}(r_{i}r_{j}),c_{2}e^{i2\pi(c_{2}^{\prime})}\mathrm{d}_{G}(r_{i}r_{j}))\), \(\mathrm{d}_{\mathcal{G}}(r_{i}r_{m})=(c_{1}e^{i2\pi(c_{1}^{\prime})}\mathrm{ d}_{G}(r_{l}r_{m}),c_{2}e^{i2\pi(c_{2}^{\prime})}\mathrm{d}_{G}(r_{l}r_{m}))\). Since \(\mathrm{d}_{G}(r_{i}r_{j})\
eq\mathrm{d}_{G}(r_{l}r_{m})\), so \(\mathrm{d}_{\mathcal{G}}(r_{i}r_{j})\
eq\mathrm{d}_{\mathcal{G}}(r_{l}r_{m})\). Thus \(\mathcal{G}\) is an edge irregular, a contradiction. Hence \(G\) is an edge regular. Conversely, let \(G\) be an edge regular. To show that \(\mathcal{G}\) is an edge regular CPFG. Consider \(\mathcal{G}\) is an edge irregular CPFG. i.e., \(\mathrm{d}_{\mathcal{G}}(r_{i}r_{j})\
eq\mathrm{d}_{\mathcal{G}}(r_{p}r_{q})\) for at least one pair of \(r_{i}r_{j},r_{p}r_{q}\in D\), \((\mathrm{d}_{\mu e^{i\alpha}}(r_{i}r_{j}),\mathrm{d}_{vv^{i\beta}}(r_{i}r_{j} ))\
eq(\mathrm{d}_{\mu e^{i\alpha}}(r_{p}q),\mathrm{d}_{vv^{i\beta}}(r_{p}r_{q}))\). Now \(\mathrm{d}_{\mu e^{i\alpha}}(r_{i}r_{j})\
eq\mathrm{d}_{\mu e^{i\alpha}}(r_{p}r_ {q})\) implies  
\(\sum\limits_{ \begin{subarray}{c}r_{i}r_{k}\in D\\ k\
eq j\end{subarray} }\mu_{\mathcal{G}}(r_{k}r_{k})e^{ \begin{subarray}{c}i \sum\limits_{r_{i}r_{k}\in D\\ k\
eq i\end{subarray} }}+\sum\limits_{ \begin{subarray}{c}r_{i}r_{k}\in D\\ k\
eq i\end{subarray} }\mu_{\mathcal{G}}(r_{j}r_{k})e^{ \begin{subarray}{c}i \sum\limits_{r_{i}r_{k}\in D}\alpha_{\mathcal{G}}(r_{l})\\ k\
eq i\end{subarray} }\
eq\sum\limits_{ \begin{subarray}{c}r_{i}r_{k}\in D\\ k\
eq i\end{subarray} }\mu_{\mathcal{G}}(r_{p}r_{s})e^{ \begin{subarray}{c}i \sum\limits_{r_{i}r_{k}\in D}\alpha_{\mathcal{G}}(r_{p}r_{s})\\ k\
eq i\end{subarray} }\)  
\(+\sum\limits_{ \begin{subarray}{c}r_{i}r_{j}\in D\\ k\
eq i\end{subarray} }\mu_{\mathcal{G}}(r_{q}r_{q})e^{ \begin{subarray}{c}i \sum\limits_{r_{i}r_{k}\in D}\beta_{\mathcal{G}}(r_{r_{q}})\\ k\
eq p\end{subarray} }\), since \(\mu_{\mathcal{G}}e^{i\alpha_{\mathcal{G}}}\) is a constant, so \(\mathrm{d}_{G}(r_{i}r_{j})\
eq\mathrm{d}_{G}(r_{p}r_{q})\), a contradiction. Therefore \(\mathcal{G}\) is an edge regular CPFG. Theorem 4.1: _Let \(\mathcal{G}=(\mathcal{C},\mathcal{D})\) be a regular CPFG. Then \(\mathcal{G}\) is an edge regular CPFG if and only if \((\mu_{\mathcal{D}}e^{i\alpha_{\mathcal{D}}},\
u_{\mathcal{D}}e^{i\beta_{ \mathcal{D}}})\) is a constant function._  
Proof: Let \(\mathcal{G}\) be a \((r_{1},r_{2})\)-regular CPFG i.e., \(\mathrm{d}_{\mathcal{G}}(r_{i})=(r_{1},r_{2})\) for all \(r_{i}\in C\). Suppose that \((\mu_{\mathcal{D}}e^{i\alpha_{\mathcal{D}}},\
u_{\mathcal{D}}e^{i\beta_{ \mathcal{D}}})\) is a constant function. Then \(\mu_{\mathcal{G}}(r_{i}r_{j})e^{i\alpha_{\mathcal{D}}(r_{i}r_{j})}=c_{1}e^{i2 \pi(c_{1}^{\prime})}\) and \(\
u_{\mathcal{D}}(r_{i}r_{j})e^{i\beta_{\mathcal{D}}(r_{i}r_{j})}=c_{2}e^{i2\pi(c_ {2}^{\prime})}\) for all \(r_{i}r_{j}\in D\), where \(c_{1}e^{i2\pi(c_{1}^{\prime})}\) and \(c_{2}e^{i2\pi(c_{2}^{\prime})}\) are constants.