## 3 Edge Regularity of a Graph in Complex Pythagorean Fuzzy Circumstances
}c_{1}e^{i2\pi(c_{1}^{\prime})}+c_{1}e^{i2\pi(c_{1}^{ \prime})}\] \[= c_{1}e^{i2\pi(c_{1}^{\prime})}(l-1)+c_{1}e^{i2\pi(c_{1}^{\prime} )}(l-1)+c_{1}e^{i2\pi(c_{1}^{\prime})}\] \[= c_{1}e^{i2\pi(c_{1}^{\prime})}(2l-1)\text{ for all }r_{i}r_{j}\in D.\]  
Similarly, \(t\mathrm{d}_{\
u e^{\mathrm{i}\beta}}(r_{i}r_{j})=c_{2}e^{i2\pi(c_{2}^{\prime })}(2l-1)\) for all \(r_{i}r_{j}\in D\). Therefore \(\mathcal{G}\) is a totally edge regular CPFG. Conversely, assume that \(\mathcal{G}\) is regular as well as totally edge regular CPFG. We show that \((\mu_{\mathcal{D}}e^{i\alpha_{\mathcal{D}}},\
u_{\mathcal{D}}e^{i\beta_{ \mathcal{D}}})\) is a constant function. Since CPFG \(\mathcal{G}\) is regular, \(\mathrm{d}_{\mathcal{G}}(r_{i})=(r_{1},r_{2})\) for all \(r_{i}\in C\). Also, \(\mathcal{G}\) is a totally edge regular, \(t\mathrm{d}_{\mathcal{G}}(r_{i}r_{j})=(\mathcal{S}_{1},\mathcal{S}_{2})\) for all \(r_{i}r_{j}\in D\). According to the definition of total degree of an edge, we have \(t\mathrm{d}_{\mathcal{G}}(r_{i}r_{j})=(t\mathrm{d}_{\mu e^{\mathrm{i}\alpha}}(r _{i}r_{j}),t\mathrm{d}_{\
u e^{\mathrm{i}\beta}}(r_{i}r_{j}))\), where  
\[t\mathrm{d}_{\mu e^{\mathrm{i}\alpha}}(r_{i}r_{j}) = \mathrm{d}_{\mu e^{\mathrm{i}\alpha}}(r_{i})+\mathrm{d}_{\mu e^{ \mathrm{i}\alpha}}(r_{j})-\mu_{\mathcal{G}}(r_{i}r_{j})e^{i\alpha_{\mathcal{G} }(r_{i}r_{j})}\] \[\mathcal{S}_{1} = r_{1}+r_{1}-\mu_{\mathcal{D}}(r_{i}r_{j})e^{i\alpha_{\mathcal{G} }(r_{i}r_{j})}\] \[\mu_{\mathcal{D}}(r_{i}r_{j}) = 2r_{1}-\mathcal{S}_{1}\text{ for all }r_{i}r_{j}\in D.\] Similarly, \(\
u_{\mathcal{D}}(r_{i}r_{j})=2r_{2}-\mathcal{S}_{2}\) for all \(r_{i}r_{j}\in D\). Hence \((\mu_{\mathcal{D}}e^{i\alpha_{\mathcal{D}}},\
u_{\mathcal{D}}e^{i\beta_{ \mathcal{D}}})\) is a constant function. **Definition 25**. : _A CPFG \(\mathcal{G}\) on \(G\) is said to be partially edge regular, if \(G\) is an edge regular._  
**Definition 26**. : _A CPFG \(\mathcal{G}\) on \(G\) is said to be full edge regular, if \(\mathcal{G}\) is edge regular as well as partially edge regular._  
**Theorem 14**. : _Let \(\mathcal{G}=(\mathcal{C},\mathcal{D})\) be a CPFG on \(G\) such that \((\mu_{\mathcal{D}}e^{i\alpha_{\mathcal{D}}},\
u_{\mathcal{D}}e^{i\beta_{ \mathcal{D}}})\) is a constant function. If \(\mathcal{G}\) is full regular CPFG, then \(\mathcal{G}\) is full edge regular CPFG._  
Proof. : Suppose that \((\mu_{\mathcal{D}}e^{i\alpha_{\mathcal{D}}},\
u_{\mathcal{D}}e^{i\beta_{ \mathcal{D}}})\) is a constant function. \(\mu_{\mathcal{D}}(r_{i}r_{j})e^{i\alpha_{\mathcal{D}}(r_{i}r_{j})}=c_{1}e^{i2 \pi(c_{1}^{\prime})}\) 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. Let \(\mathcal{G}\) be a full regular CPFG, then \(\mathrm{d}_{G}(r_{i})=l\) and \(\mathrm{d}_{\mathcal{G}}(r_{i})=(r_{1},r_{2})\) for all \(r_{i}\in C\), where \(l,r_{1}\) and \(r_{2}\) are constants. Therefore \(\mathrm{d}_{G}(r_{i}r_{j})=\mathrm{d}_{G}(r_{i})+\mathrm{d}_{G}(r_{j})-2=2l-2=\) constant. Hence the graph \(G\) is an edge regular. Now, \(\mathrm{d}_{\mathcal{G}}(r_{i}r_{j})=(\mathrm{d}_{\mu e^{\mathrm{i}\alpha}}(r_{i} r_{j}),\mathrm{d}_{\
u e^{\mathrm{i}\beta}}(r_{i}r_{j}))\) for all \(r_{i}r_{j}\in D\), where  
\[\mathrm{d}_{\mu e^{\mathrm{i}\alpha}}(r_{i}r_{j}) = \mathrm{d}_{\mu e^{\mathrm{i}\alpha}}(r_{i})+\mathrm{d}_{\mu e^{ \mathrm{i}\alpha}}(r_{j})-2\mu_{\mathcal{D}}(r_{i}r_{j})e^{2i\alpha_{\mathcal{D} }(r_{i}r_{j})}\] \[= r_{1}+r_{1}-2c_{1}e^{i2\pi(c_{1}^{\prime})}\] \[= 2r_{1}-2c_{1}e^{i2\pi(c_{1}^{\prime})}=\text{constant}.\]  
Similarly, \(\mathrm{d}_{\
u e^{\mathrm{i}\beta}}(r_{i}r_{j})=2r_{2}-2c_{2}e^{i2\pi(c_{2}^{ \prime})}=\)constant, for all \(r_{i}r_{j}\in D\). Hence \(\mathcal{G}\) is an edge regular CPFG. Thus \(\mathcal{G}\) is full edge regular CPFG. **Theorem 15**.