## 1 Introduction
[27] introduced the concept of complex intuitionistic fuzzy graph with its application in cellular network provider companies. Further, [PERSON] and [PERSON] [28] extended the concept of complex fuzzy graphs to complex neutrosophic graphs. As CPFS is a more generalized version of the existing theories such as FSs, IFSs, CFSs and CIFSs. Thus, motivated by this, in this paper, within complex Pythagorean fuzzy contexts, we introduce the innovative concept of complex Pythagorean fuzzy graphs (CPFGs) in which pairs of the membership degrees represent the two-dimensional information. We develop operations on two CPFGs and investigate their desirable properties. We define the concepts of regular and edge regular graphs with appropriate illustration and examine some of their crucial properties with complex Pythagorean fuzzy information. Aggregation operators have great importance in many fields of information processing such as decision making, medical diagnosis, pattern recognition, data mining machine retrieval and machine learning, and so forth. Aggregation operators are commonly used to convert all the inputted individual information into a single value. So, we also develop systematic operations and aggregation operators to aggregate complex Pythagorean fuzzy information. Finally, we develop a CPFG based MADM approach to handle situations in which the attributes graphic structure is uncertain. The paper is structured as follows: Section 2 proposes a new generalization of Pythagorean fuzzy graphs--called CPFG--and investigates its properties in detail. Section 3 discusses the edge regularity of a graph in complex Pythagorean fuzzy circumstances. In Section 4, we discuss the aggregation operators of CPFSs and provide the application of CPFSs and CPFGs in MADM and finally we draw conclusions and elaborate on future work in Section 5. **Definition 1** ([15]). : _Let \(Y\) be the universe of discourse. A complex Pythagorean fuzzy set \(\mathscr{C}\) defined on \(Y\) is an object of the form_  
\[\mathscr{C}=\{(r,\mu_{\mathscr{C}}(r)e^{i\alpha_{\mathscr{C}}(r)},\
u_{\mathscr{ C}}(r)e^{i\beta_{\mathscr{C}}(r)}):r\in Y\},\]  
_where \(i=\sqrt{-1},\mu_{\mathscr{C}}(r),\
u_{\mathscr{C}}(r)\in[0,1]\), \(\alpha_{\mathscr{C}}(r),\beta_{\mathscr{C}}(r)\in[0,2\pi]\), \(0\leq\mu_{\mathscr{C}}^{2}(r)+\
u_{\mathscr{C}}^{2}(r)\leq 1\) and \(0\leq\alpha_{\mathscr{C}}^{2}(r)+\beta_{\mathscr{C}}^{2}(r)\leq 2\pi\)._  
**Definition 2** ([15]). : _Let \(\mathscr{C}=\{(r,\mu_{\mathscr{C}}(r)e^{i\alpha_{\mathscr{C}}(r)},\
u_{\mathscr{ C}}(r)e^{i\beta_{\mathscr{C}}(r)}):r\in Y\}\), \(\mathscr{C}_{1}=\{(r,\mu_{\mathscr{C}_{1}}(r)e^{i\alpha_{\mathscr{C}_{1}}(r)}, \
u_{\mathscr{C}_{1}}(r)e^{i\beta_{\mathscr{C}_{1}}(r)}):r\in Y\}\), and \(\mathscr{C}_{2}=\{(r,\mu_{\mathscr{C}_{2}}(r)e^{i\alpha_{\mathscr{C}_{2}}(r)}, \
u_{\mathscr{C}_{2}}(r)e^{i\beta_{\mathscr{C}_{2}}(r)}):r\in Y\}\) be the three CPFSs in \(Y\), then_  
**(i)**: \(\mathscr{C}_{1}\subseteq\mathscr{C}_{2}\) _if and only if_ \(\mu_{\mathscr{C}_{1}}(r)\leq\mu_{\mathscr{C}_{2}}(r),\
u_{\mathscr{C}_{1}}(r) \geq\
u_{\mathscr{C}_{2}}(r)\) _for amplitude terms and_ \(\alpha_{\mathscr{C}_{1}}(r)\leq\alpha_{\mathscr{C}_{2}}(r)\)_,_ \(\beta_{\mathscr{C}_{1}}(r)\geq\beta_{\mathscr{C}_{2}}(r)\) _for phase terms, for all_ \(r\in Y\)_;_  
**(ii)**: \(\mathscr{C}_{1}=\mathscr{C}_{2}\) _if and only if_ \(\mu_{\mathscr{C}_{1}}(r)=\mu_{\mathscr{C}_{2}}(r),\
u_{\mathscr{C}_{1}}(r)=\
u_{ \mathscr{C}_{2}}(r)\) _for amplitude terms and_ \(\alpha_{\mathscr{C}_{1}}(r)=\alpha_{\mathscr{C}_{2}}(r)\)_,_ \(\beta_{\mathscr{C}_{1}}(r)=\beta_{\mathscr{C}_{2}}(r)\) _for phase terms, for all_ \(r\in Y\)_;_
**(iii)**: \(\mathscr{C}=\{(r,\
u_{\mathscr{C}}(r)e^{i\beta_{\mathscr{C}}(r)},\mu_{ \mathscr{C}}(r)e^{i\alpha_{\mathscr{C}}(r)}):r\in Y\}\)_._  
For simplicity, the pair \((\mu e^{i\alpha},\
u e^{i\beta})\) is called the complex Pythagorean fuzzy number (CPFN), where \(\mu,\
u\in[0,1]\) such that \(\mu^{2}+\
u^{2}\leq 1\) and \(\alpha,\beta\in[0,2\pi]\) such that \(0\leq\alpha^{2}+\beta^{2}\leq 2\pi\). **Example 1**.