## 1 Introduction
\begin{array}{ll}\mu_{\mathcal{ C}_{1}}(r)e^{i\alpha_{\mathcal{G}_{1}}(r)}&\mbox{if $r\in C_{1}-C_{2}$},\\ \mu_{\mathcal{G}_{2}}(r)e^{i\alpha_{\mathcal{G}_{2}}(r)}&\mbox{if $r\in C_{2}-C_{1}$},\\ (\mu_{\mathcal{G}_{1}}(r)\vee\mu_{\mathcal{G}_{2}}(r))e^{i(\alpha_{\mathcal{G}_ {1}}(r)\vee\alpha_{\mathcal{G}_{2}}(r))}&\mbox{if $r\in C_{1}\cap C_{2}$},\\ \end{array} \right.\)
**(ii)**: \((\
u_{\mathcal{G}_{1}}\cup\
u_{\mathcal{G}_{2}})(r)e^{i(\beta_{\mathcal{G}_ {1}}\cup\beta_{\mathcal{G}_{2}})(r)}=\left\{ \begin{array}{ll}\
u_{\mathcal{ G}_{1}}(r)e^{i\beta_{\mathcal{G}_{1}}(r)}&\mbox{if $r\in C_{1}-C_{2}$},\\ \
u_{\mathcal{G}_{2}}(r)e^{i\widehat{\beta}_{\mathcal{G}_{2}}(r)}&\mbox{if $r\in C_{2}-C_{1}$},\\ (\
u_{\mathcal{G}_{1}}(r)\vee\
u_{\mathcal{G}_{2}}(r))e^{i(\beta_{\mathcal{G} _{1}}(r)\wedge\beta_{\mathcal{G}_{2}}(r))}&\mbox{if $r\in C_{1}\cap C_{2}$},\\ \end{array} \right.\)
**(iii)**: \((\mu_{\mathcal{G}_{1}}\cup\mu_{\mathcal{G}_{2}})(rs)e^{i(\alpha_{\mathcal{G}_ {1}}\cup\alpha_{\mathcal{G}_{2}})(rs)}=\left\{ \begin{array}{ll}\mu_{\mathcal{ G}_{1}}(rs)e^{i\alpha_{\mathcal{G}_{1}}(rs)}&\mbox{if $rs\in D_{1}-D_{2}$},\\ \mu_{\mathcal{G}_{2}}(rs)e^{i\alpha_{\mathcal{G}_{2}}(rs)}&\mbox{if $rs\in D_{2}-D_{1}$},\\ (\mu_{\mathcal{G}_{1}}(rs)\vee\mu_{\mathcal{G}_{2}}(rs))e^{i(\alpha_{\mathcal{G }_{1}}(rs)\vee\alpha_{\mathcal{G}_{2}}(rs))}&\mbox{if $rs\in D_{1}\cap D_{2}$},\\ \end{array} \right.\)
**(iv)**: \((\
u_{\mathcal{G}_{1}}\cup\
u_{\mathcal{G}_{2}})(rs)e^{i(\beta_{\mathcal{G} _{1}}\cup\beta_{\mathcal{G}_{2}})(rs)}=\left\{ \begin{array}{ll}\
u_{\mathcal{ G}_{1}}(rs)e^{i\beta_{\mathcal{G}_{1}}(rs)}&\mbox{if $rs\in D_{1}-D_{2}$},\\ \
u_{\mathcal{G}_{2}}(rs)e^{i\widehat{\beta}_{\mathcal{G}_{2}}(rs)}&\mbox{if $rs\in D_{2}-D_{1}$},\\ (\
u_{\mathcal{G}_{1}}(rs)\wedge\
u_{\mathcal{G}_{2}}(rs))e^{i(\beta_{\mathcal{G }_{1}}(rs)\wedge\beta_{\mathcal{G}_{2}}(rs))}&\mbox{if $rs\in D_{1}\cap D_{2}$}.\end{array} \right.\)  
**Theorem 2**. : _The union \(\mathcal{G}_{1}\cup\mathcal{G}_{2}\) of \(\mathcal{G}_{1}\) and \(\mathcal{G}_{2}\) is a CPFG of \(G_{1}\cup G_{2}\) if and only if \(\mathcal{G}_{1}\) and \(\mathcal{G}_{2}\) are CPFGs of \(G_{1}\) and \(G_{2}\), respectively, where \(C_{1}\cap C_{2}=\emptyset\)._  
**Definition 9**. : _The ring-sum \(\mathcal{G}_{1}\oplus\mathcal{G}_{2}=(\mathcal{C}_{1}\oplus\mathcal{C}_{2}, \mathcal{G}_{1}\oplus\mathcal{G}_{2})\) 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 \(G_{1}\) and \(G_{2}\), respectively, is defined as follows:_  
\[(\mu_{\mathcal{C}_{1}}\oplus\mu_{\mathcal{C}_{2}})(r)e^{i(\alpha_{ \mathcal{G}_{1}}\oplus\alpha_{\mathcal{G}_{2}})(r)}=(\mu_{\mathcal{C}_{1}}\cup \mu_{\mathcal{C}_{2}})(r)e^{i(\alpha_{\mathcal{C}_{1}}\cup\alpha_{\mathcal{G}_{ 2}})(r)},\] \[(\,\,\
u_{\mathcal{G}_{1}}\oplus\
u_{\mathcal{G}_{2}})(r)e^{i(\, \,\beta_{\mathcal{G}_{1}}\oplus\beta_{\mathcal{G}_{2}})(r)}=(\,\,\
u_{\mathcal{ C}_{1}}\cup\
u_{\mathcal{G}_{2}})(r)e^{i(\,\,\beta_{\mathcal{C}_{1}}\,\,\beta_{ \mathcal{C}_{2}})(r)}\,\,\,\,\mbox{if $r\in C_{1}\cup C_{2}$},\]  
\[(\mu_{\mathcal{G}_{1}}\oplus\mu_{\mathcal{G}_{2}})(rs)e^{i(\alpha_{ \mathcal{G}_{1}}\oplus\alpha_{\mathcal{G}_{2}})(rs)} = \left\{ \begin{array}{ll}\mu_{\mathcal{G}_{1}}(rs)e^{i\alpha_{\mathcal{ G}_{1}}(rs)}&\mbox{if $rs\in D_{1}-D_{2}$},\\ \mu_{\mathcal{G}_{2}}(rs)e^{i\alpha_{\mathcal{G}_{2}}(rs)}&\mbox{if $rs\in D_{2}-D_{1}$},\\ 0&\mbox{if $rs\in D_{1}\cap D_{2}$}.\end{array} \right.\] \[(\,\,\
u_{\mathcal{G}_{1}}\oplus\
u_{\mathcal{G}_{2}})(rs)e^{i(\,\, \beta_{\mathcal{G}_{1}}\oplus\beta_{\mathcal{G}_{2}})(rs)} = \left\{ \begin{array}{ll}\
u_{\mathcal{G}_{1}}(rs)e^{i\beta_{\mathcal{G }_{1}}(rs)}&\mbox{if $rs\in D_{1}-D_{2}$},\\ \
u_{\mathcal{G}_{2}}(rs)e^{i\beta_{\mathcal{G}_{2}}(rs)}&\mbox{if $rs\in D_{2}-D_{1}$},\\ 0&\mbox{if $rs\in D_{1}\cap D_{2}$}.\end{array} \right.\]  
**Proposition 1**.