## 1 Introduction
: _If \(\mathcal{G}_{1}=(\mathcal{C}_{1},\mathcal{G}_{1})\) and \(\mathcal{G}_{2}=(\mathcal{C}_{2},\mathcal{G}_{2})\) are the CPFGs, then \(\mathcal{G}_{1}\oplus\mathcal{G}_{2}\) is the CPFG._  
**Definition 10**. : _Let \(\mathcal{G}_{1}\) and \(\mathcal{G}_{2}\) be two CPFGs of \(G_{1}\) and \(G_{2}\), respectively. The join \(\mathcal{G}_{1}+\mathcal{G}_{2}=(\mathcal{C}_{1}+\mathcal{C}_{2  
**(iii)**: \(\left\{ \begin{array}{ll}(\mu_{\mathcal{D}_{1}}+\mu_{\mathcal{D}_{2}})(rs)e^{i( \alpha_{\mathcal{D}_{1}}+\alpha_{\mathcal{D}_{2}})(rs)}=\mu_{\mathcal{C}_{1}}( r)\wedge\mu_{\mathcal{C}_{2}}(s)e^{i\alpha_{\mathcal{C}_{1}}(r)\wedge\alpha_{ \mathcal{C}_{2}}(s)}\\ (\,v_{\mathcal{D}_{1}}+\,v_{\mathcal{D}_{2}})(rs)e^{i(\,\beta_{\mathcal{D}_{1 }}+\,\beta_{\mathcal{D}_{2}})(rs)}=\,v_{\mathcal{C}_{1}}(r)\vee\,v_{\mathcal{ C}_{2}}(s)e^{i\beta_{\mathcal{C}_{1}}(r)\vee\,\beta_{\mathcal{C}_{2}}(s)}& \text{ if }rs\in D^{\prime},\end{array} \right.\)  
_where \(D^{\prime}\) is the set of all edges joining the vertices of \(C_{1}\) and \(C_{2}\), \(C_{1}\cap C_{2}=\emptyset\)._  
**Theorem 3**. : _The join \(\mathcal{G}_{1}+\mathcal{G}_{2}\) of \(\mathcal{G}_{1}\) and \(\mathcal{G}_{2}\) is a CPFG of \(G_{1}+G_{2}\) 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 11**. : _The degree of a vertex \(r\in C\) in a CPFG \(\mathcal{G}\) is denoted by \(\mathrm{d}_{\mathcal{G}}(r)\), and is defined as \(\mathrm{d}_{\mathcal{G}}(r)=(\mathrm{d}_{\mu^{\mathrm{kin}}}(r),\mathrm{d}_{ vv^{\beta}}(r))\), where_  
\[\mathrm{d}_{\mu^{\mathrm{kin}}}(r) = \sum_{r,s\
eq r\in C}\mu_{\mathcal{G}}(rs)e^{i\sum_{r\in C} \alpha_{\mathcal{G}}(rs)},\] \[\mathrm{d}_{vv^{\beta}}(r) = \sum_{r,s\
eq r\in C}\
u_{\mathcal{G}}(rs)e^{i\sum_{r\in C} \beta_{\mathcal{G}}(rs)}.\]  
**Definition 12**. : _The total degree of a vertex \(r\in C\) in a CPFG \(\mathcal{G}\) is denoted by \(\mathrm{td}_{\mathcal{G}}(r)\), and is defined as \(\mathrm{td}_{\mathcal{G}}(r)=(\mathrm{td}_{\mu^{\mathrm{kin}}}(r),\mathrm{td} _{v}e^{i\beta}(r))\), where_  
\[\mathrm{td}_{\mu^{\mathrm{kin}}}(r) = \sum_{r,s\
eq r\in C}\mu_{\mathcal{G}}(rs)e^{i\sum_{r\in C} \mu_{\mathcal{G}}(rs)}+\mu_{\mathcal{G}}(r)e^{i\alpha_{\mathcal{G}}(r)},\] \[\mathrm{td}_{v}e^{i\beta}(r) = \sum_{r,s\
eq r\in C}\
u_{\mathcal{G}}(rs)e^{i\sum_{r\in C}\
u_{ \mathcal{G}}(rs)}+\
u_{\mathcal{G}}(r)e^{i\beta_{\mathcal{G}}(r)}.\]  
**Definition 13**. : _Let \(\mathcal{G}_{1}\) and \(\mathcal{G}_{2}\) be two CPFGs. For any vertex \(r\in C_{1}\cup C_{2}\), there are three cases to consider._  
_Case 1_: _Either_ \(r\in C_{1}-C_{2}\) _or_ \(r\in C_{2}-C_{1}\)_. Then, no edge incident at_ \(r\) _lies in_ \(D_{1}\cap D_{2}\)_. Thus, for_ \(r\in C_{1}-C_{2}\)_,_  
\[(d_{\mu^{\mathrm{kin}}})_{\mathcal{G}_{1}\cup\mathcal{G}_{2}}(r)=\sum_{r\in D_ {1}}\mu_{\mathcal{G}_{1}}(rs)e^{i\sum\limits_{r\in D_{1}}\alpha_{\mathcal{G}_{ 1}}(rs)}=(d_{\mu^{\mathrm{kin}}})_{\mathcal{G}_{1}}(r),\]  
\[(d_{v^{\beta}})_{\mathcal{G}_{1}\cup\mathcal{G}_{2}}(r)=\sum_{r\in D_{1}}\
u_{ \mathcal{G}_{1}}(rs)e^{i\sum\limits_{r\in D_{1}}\beta_{\mathcal{G}_{1}}(rs)}=( d_{v^{\beta}})_{\mathcal{G}_{1}}(r),\]  
\[(td_{\mu^{\mathrm{kin}}})_{\mathcal{G}_{1}\cup\mathcal{G}_{2}}(r)=(td_{\mu^{ \mathrm{kin}}})_{\mathcal{G}_{1}}(r),(td_{v^{\beta}})_{\mathcal{G}_{1}\cup \mathcal{G}_{2}}(r)=(td_{v^{\beta}})_{\mathcal{G}_{1}}(r).\]  
_For_ \(r\in C_{2}-C_{1}\)_,_  
\[(d_{\mu^{\mathrm{kin}}})_{\mathcal{G}_{1}\cup\mathcal{G}_{2}}(r)=\sum_{r\in D_ {2}}\mu_{\mathcal{G}_{2}}(rs)e^{i\sum\limits_{r\in D_{2}}\alpha_{\mathcal{G}_{ 2}}(rs)}=(d_{\mu^{\mathrm{kin}}})_{\mathcal{G}_{2}}(r),\]  
\[(d_{v^{\beta}})_{\mathcal{G}_{1}\cup\mathcal{G}_{2}}(r)=\sum_{r\in D_{2}}\
u_{ \mathcal{G}_{2}}(rs)e^{i\sum\limits_{r\in D_{2}}\beta_{\mathcal{G}_{2}}(rs)}=( d_{v^{\beta}})_{\mathcal{G}_{2}}(r).\]  
_Case 2_: \(r\in C_{1}\cap C_{2}\) _but no edge incident at_ \(r\) _lies in_ \(D_{1}\cap D_{2}\)_.