## 1 Introduction
: _Suppose a fixed set \(Y\) contains only one element \(r\), \(\mu_{\mathscr{C}}(r)=0.4,\alpha_{\mathscr{C}}(r)=1.2\pi,\
u_{\mathscr{C}}(r)= 0.7,\beta_{\mathscr{C}}(r)=1.6\pi\). Then \(\mathscr{C}=\{(r,0.4e^{i2\pi(0.6)},0.7e^{i2\pi(0.8)})\}\) is a CPFN and denoted by \(\mathscr{C}=(0.4e^{i2\pi(0.6)},0.7e^{i2\pi(0.8)})\) for simplicity._  
**2. Graphs in Complex Pythagorean Fuzzy Environment**  
In this section, the innovative concepts of complex Pythagorean fuzzy relation and CPFG are introduced and some related properties are investigated. **Definition 3**. : _Let \(\mathscr{C}_{1}=\{(r,\mu_{\mathscr{C}_{1}}(r)e^{i\alpha_{\mathscr{C}_{1}}(r)},\
u_{\mathscr{C}_{1}}(r)e^{i\beta_{\mathscr{C}_{1}}(r)})\quad:\quad r\in Y\}\), and \(\mathscr{C}_{2}=\{(r,\mu_{\mathscr{C}_{2}}(r)e^{i\mu_{\mathscr{C}_{2}}(r)},\
u_ {\mathscr{C}_{2}}(r)e^{i\beta_{\mathscr{C}_{2}}(r)}):r\in Y\}\) be two CPFSs in \(Y\), then_  
**(i)**: \(\mathscr{C}_{1}\cup\mathscr{C}_{2}=\Big{\{}\Big{(}r,(\mu_{\mathscr{C}_{1}}(r) \vee\mu_{\mathscr{C}_{2}}(r))e^{i(\alpha_{\mathscr{C}_{1}}(r)\vee\alpha_{ \mathscr{C}_{2}}(r))},(\
u_{\mathscr{C}_{1}}(r)\wedge\
u_{\mathscr{C}_{2}}(r) )e^{i(\beta_{\mathscr{C}_{1}}(r)\wedge\beta_{\mathscr{C}_{2}}(r))}\Big{)}:r \in Y\Big{\}}\)_;_
**(ii)**: \(\mathscr{C}_{1}\cap\mathscr{C}_{2}=\Big{\{}\Big{(}r,(\mu_{\mathscr{C}_{1}}(r) \wedge\mu_{\mathscr{C}_{2}}(r))e^{i(\alpha_{\mathscr{C}_{1}}(r)\wedge\alpha_{ \mathscr{C}_{2}}(r))},(\
u_{\mathscr{C}_{1}}(r)\vee\
u_{\mathscr{C}_{2}}(r))e ^{i(\beta_{\mathscr{C}_{1}}(r)\vee\beta_{\mathscr{C}_{2}}(r))}\Big{)}:r\in Y \Big{\}}\)_._  
**Definition 4**. : _A CPFS \(\mathscr{D}\) in \(Y\times Y\) is said to be a complex Pythagorean fuzzy relation in \(Y\), characterized by_  
\[\mathscr{D}=\big{\{}\langle rs,\mu_{\mathscr{D}}(rs)e^{i\alpha_{\mathscr{D}}( rs)},\
u_{\mathscr{D}}(rs)e^{i\beta_{\mathscr{D}}(rs)}\rangle\mid rs\in Y\times Y \big{\}},\]  
_where \(\mu_{\mathscr{D}}:Y\times Y\to[0,1]\) and \(\
u_{\mathscr{D}}:Y\times Y\to[0,1]\) depict the membership and non-membership function of \(\mathscr{D}\), respectively, such that \(0\leq\mu_{\mathscr{D}}^{2}(rs)+\
u_{\mathscr{D}}^{2}(rs)\leq 1\) and \(0\leq\alpha_{\mathscr{D}}^{2}(rs)+\beta_{\mathscr{D}}^{2}(rs)\leq 2\pi\) for all \(rs\in Y\times Y\)._  
**Definition 5**. : _A complex Pythagorean fuzzy graph on a non-empty set \(Y\) is a pair \(\mathscr{G}=(\mathscr{C},\mathscr{D})\), where \(\mathscr{C}\) is a complex Pythagorean fuzzy set on \(Y\) and \(\mathscr{D}\) is a complex Pythagorean fuzzy relation on \(Y\) such that:_  
\[\mu_{\mathscr{D}}(rs)e^{i\alpha_{\mathscr{D}}(rs)} \leq(\mu_{\mathscr{C}}(r)\wedge\mu_{\mathscr{C}}(s))e^{i(\alpha_{ \mathscr{C}}(r)\wedge\alpha_{\mathscr{C}}(s))},\] \[\
u_{\mathscr{D}}(rs)e^{i\beta_{\mathscr{D}}(rs)} \leq(\
u_{\mathscr{C}}(r)\vee\
u_{\mathscr{C}}(s))e^{i(\beta_{ \mathscr{C}}(r)\vee\beta_{\mathscr{C}}(s))},\]  
\(0\leq\mu_{\mathscr{D}}^{2}(rs)+\
u_{\mathscr{D}}^{2}(rs)\leq 1\) _and \(0\leq\alpha_{\mathscr{D}}^{2}(rs)+\beta_{\mathscr{D}}^{2}(rs)\leq 2\pi\) for all \(r,s\in Y\). We call \(\mathscr{C}\) and \(\mathscr{D}\) the complex Pythagorean fuzzy vertex set and the complex Pythagorean fuzzy edge set of \(\mathscr{G}\), respectively._  
**Example 2**. : _Consider a graph \(G=(C,D)\), where \(C=\{s_{1},s_{2},s_{3},s_{4},s_{5}\}\) is the vertex set and \(D=\{s_{1}s_{2},s_{2}s_{3},s_{3}s_{4},s_{1}s_{5},s_{2}s_{5},s_{3}s_{5},s_{4}s_{5}\}\) is the edge set of \(G\).