## 1 Introduction
Then, by utilizing Definition 7, we have  
\[\overline{\overline{\mu_{\mathcal{G}}}(r)}e^{i\overline{\mu_{\mathcal{G}}(r)}}= \overline{\mu_{\mathcal{G}}(r)}e^{i\overline{\alpha_{\mathcal{G}}(r)}}=\mu_{ \mathcal{G}}(r)e^{i\alpha_{\mathcal{G}}(r)}\text{ and }\overline{\overline{v_{\mathcal{G}}(r)}}e^{i \overline{\beta_{\mathcal{G}}(r)}}=\overline{v_{\mathcal{G}}(r)}e^{i\overline {\beta_{\mathcal{G}}(r)}}=\
u_{\mathcal{G}}(r)e^{i\beta_{\mathcal{G}}(r)}\text { for all }r\in C.\]  
If \(\mu_{\mathcal{G}}(rs)e^{i\mu_{\mathcal{G}}(rs)}=\
u_{\mathcal{G}}(rs)e^{i\beta_ {\mathcal{G}}(rs)}=0\), then  
\[\overline{\overline{\mu_{\mathcal{G}}(rs)}}e^{i\overline{\mu_{\mathcal{G}}( rs)}}=(\overline{\mu_{\mathcal{G}}(r)}\wedge\overline{\mu_{\mathcal{G}}(s)})e^{i (\overline{\alpha_{\mathcal{G}}(r)}\wedge\overline{\alpha_{\mathcal{G}}(s)} )}=(\mu_{\mathcal{G}}(r)\wedge\mu_{\mathcal{G}}(s))e^{i(\alpha_{\mathcal{G}}( r)\wedge\alpha_{\mathcal{G}}(s))}=\mu_{\mathcal{G}}(rs)e^{i\alpha_{ \mathcal{G}}(rs)},\]  
\[\overline{\overline{\
u_{\mathcal{G}}(rs)}}e^{i\overline{\beta_{\mathcal{G}}( rs)}}=(\overline{\
u_{\mathcal{G}}(r)}\vee\overline{\
u_{\mathcal{G}}(s)})e^{i (\overline{\beta_{\mathcal{G}}(r)}\vee\overline{\beta_{\mathcal{G}}(s)})}=( \
u_{\mathcal{G}}(r)\vee\
u_{\mathcal{G}}(s))e^{i(\beta_{\mathcal{G}}(r) \vee\beta_{\mathcal{G}}(s))}=\
u_{\mathcal{G}}(rs)e^{i\beta_{\mathcal{G}}(rs)}.\]  
If \(0<\mu_{\mathcal{G}}(rs)e^{i\alpha_{\mathcal{G}}(rs)},\
u_{\mathcal{G}}(rs)e^{i \beta_{\mathcal{G}}(rs)}\leq 1\), then  
\[\overline{\overline{\mu_{\mathcal{G}}(rs)}}e^{i\overline{\alpha_{ \mathcal{G}}(rs)}} = (\overline{\mu_{\mathcal{G}}(r)}\wedge\overline{\mu_{\mathcal{G}}( s)})e^{i(\overline{\alpha_{\mathcal{G}}(r)}\wedge\overline{\alpha_{\mathcal{G}}(s)} )}-\overline{\mu_{\mathcal{G}}(rs)}e^{i\overline{\alpha_{\mathcal{G}}(rs)}}\] \[= (\mu_{\mathcal{G}}(r)\wedge\mu_{\mathcal{G}}(s))e^{i(\alpha_{ \mathcal{G}}(r)\wedge\alpha_{\mathcal{G}}(s))}-\big{(}(\mu_{\mathcal{G}}(r) \wedge\mu_{\mathcal{G}}(s))e^{i(\alpha_{\mathcal{G}}(r))\wedge\alpha_{ \mathcal{G}}(s))}-\mu_{\mathcal{G}}(rs)e^{i\alpha_{\mathcal{G}}(rs)}\big{)}\] \[= \mu_{\mathcal{G}}(rs)e^{i\alpha_{\mathcal{G}}(rs)},\]  
\[\mathcal{D} = \left\langle\left(\frac{s_{1}s_{4}}{0.2e^{i2\pi(0.4)}},\frac{s_{2 }s_{4}}{0.8e^{i2\pi(0.7)}},\frac{s_{3}s_{4}}{0.3e^{i2\pi(0.2)}}\right),\left( \frac{s_{1}s_{4}}{0.3e^{i2\pi(0.5)}},\frac{s_{2}s_{4}}{0.3e^{i2\pi(0.5)}}, \frac{s_{3}s_{4}}{0.4e^{i2\pi(0.3)}}\right)\right\rangle.\]0 Complement of a complex Pythagorean fuzzy graph. \[\mathcal{D} = \left\langle\left(\frac{s_{1}s_{4}}{0.2e^{i2\pi(0.4)}},\frac{s_{2 }s_{4}}{0.8e^{i2\pi(0.7)}},\frac{s_{3}s_{4}}{0.3e^{i2\pi(0.2)}}\right),\left( \frac{s_{1}s_{4}}{0.3e^{i2\pi(0.5)}},\frac{s_{2}s_{4}}{0.3e^{i2\pi(0.5)}}, \frac{s_{3}s_{4}}{0.4e^{i2\pi(0.3)}}\right)\right\rangle.\]1 \[\mathcal{D} = \left\langle\left(\frac{s_{1}s_{4}}{0.2e^{i2\pi(0.4)}},\frac{s_{2 }s_{4}}{0.8e^{i2\pi(0.7)}},\frac{s_{3}s_{4}}{0.3e^{i2\pi(0.2)}}\right),\left( \frac{s_{1}s_{4}}{0.3e^{i2\pi(0.5)}},\frac{s_{2}s_{4}}{0.3e^{i2\pi(0.5)}}, \frac{s_{3}s_{4}}{0.4e^{i2\pi(0.3)}}\right)\right\rangle.\]2 \[\mathcal{D} = \left\langle\left(\frac{s_{1}s_{4}}{0.2e^{i2\pi(0.4)}},\frac{s_{2 }s_{4}}{0.8e^{i2\pi(0.7)}},\frac{s_{3}s_{4}}{0.3e^{i2\pi(0.2)}}\right),\left( \frac{s_{1}s_{4}}{0.3e^{i2\pi(0.5)}},\frac{s_{2}s_{4}}{0.3e^{i2\pi(0.5)}}, \frac{s_{3}s_{4}}{0.4e^{i2\pi(0.3)}}\right)\right\rangle.\]3  
for all \(r,s\in C\). Hence \(\overline{\overline{\mathcal{G}}}=\mathcal{G}\). \(\qed\)  
**Definition 8**. : _The union \(\mathcal{G}_{1}\cup\mathcal{G}_{2}=(\mathcal{C}_{1}\cup\mathcal{G}_{2}, \mathcal{G}_{1}\cup\mathcal{G}_{2})\) of two CPFGs \(\mathcal{G}_{1}=(\mathcal{C}_{1},\mathcal{G}_{1})\) and \(\mathcal{G}_{2}=(\mathcal{C}_{2},\mathcal{G}_{2})\) of the graphs \(G_{1}=(C_{1},D_{1})\) and \(G_{2}=(C_{2},D_{2})\), respectively, is defined as follows:_  
**(i)**: \((\
u_{\mathcal{G}_{1}}\cup\
u_{\mathcal{G}_{2}})(r)e^{i(\alpha_{\mathcal{G}_{ 1}}\cup\alpha_{\mathcal{G}_{2}})(r)}=\left\{