# Complex Neutrosophic Hypergraphs: New Social Network Models
## 3 \(T\)-Related Complex Neutrosophic Hypergraphs
AS \(\xi^{(\alpha_{1}e^{\Theta_{1}},\beta_{1}e^{\Phi_{1}},\gamma_{1}e^{\sigma_{1}})}\in T _{r}(\mathcal{H}^{(\alpha_{1}e^{\Theta_{1}},\beta_{1}e^{\Phi_{1}},\gamma_{1}e^{ \sigma_{1}})})\), \(\xi^{(\alpha_{2}e^{\Theta_{2}},\beta_{2}e^{\Phi_{2}},\gamma_{2}e^{\sigma_{2}  
\(\delta=\xi^{(\alpha_{3}e^{\Theta_{3}},\beta_{2}e^{\Theta_{3}},\gamma_{3}e^{\pi \Lambda_{3}})}\cup\rho_{2}\cap\rho_{1}\), where \(\rho_{k}\) is an elementary CNS with support \(\tau_{k}\) and height \((\alpha_{k}e^{\Theta_{k}},\beta_{k}e^{\Theta_{k}},\gamma_{k}e^{\pi\Lambda_{k}})\), \(k=1,2\). This contradiction shows that \(\xi^{(\alpha_{2}e^{\Theta_{2}},\beta_{2}e^{\Theta_{2}},\gamma_{2}e^{\Theta_{2} })}\in T_{r}(\mathcal{H}^{(\alpha_{2}e^{\Theta_{2}},\beta_{2}e^{\Theta_{2}}, \gamma_{2}e^{\Theta_{2}})})\). Then, Lemma 1 implies that \(\xi^{(\alpha e^{\Theta_{k}},\beta e^{\Theta_{k}},\gamma_{k}e^{\pi\Lambda})}\in T _{r}(\mathcal{H}^{(\alpha e^{\Theta_{k}},\beta e^{\Theta_{k}},\gamma_{k}e^{\pi \Lambda})})\), for \(\alpha\in(\alpha_{3},\alpha_{1}]\), \(\beta\in(\beta_{3},\beta_{1}]\), \(\gamma\in(\gamma_{3},\gamma_{1}]\), \(\Theta\in(\Theta_{3},\Theta_{1}]\), \(\Phi\in(\Phi_{3},\Phi_{1}]\), \(\Psi\in(\Psi_{3},\Psi_{1}]\). Continuing the same recursive procedure, we show that \(\xi^{(\alpha e^{\Theta_{k}},\beta e^{\Theta_{k}},\gamma_{k}e^{\pi\Lambda})} \in T_{r}(\mathcal{H}^{(\alpha e^{\Theta_{k}},\beta e^{\Theta_{k}},\gamma_{k}e^ {\pi\Lambda})})\), for each \(\alpha\in(0,\alpha_{1}]\), \(\beta\in(0,\beta_{1}]\), \(\gamma\in(0,\gamma_{1}]\), \(\Theta\in(0,\Theta_{1}]\), \(\Phi\in(0,\Phi_{1}]\), \(\Psi\in(0,\Psi_{1}]\). **Example 5**: _Let \(\mathcal{H}=(\mathcal{N},\lambda)\) be a CNHG represented by the incidence matrix as given in Table 1. Please note that_  
\[\lambda^{(0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi})} = \{\{j_{1},j_{2}\},\{j_{1},j_{3}\},\{j_{2},j_{3}\}\},\] \[\lambda^{(0.6e^{(0.6)2\pi},0.6e^{(0.6)2\pi},0.6e^{(0.6)2\pi})} = \{\{j_{1},j_{2},j_{4}\},\{j_{1},j_{3},j_{4}\},\{j_{2},j_{3}\}\},\] \[\lambda^{(0.3e^{(0.3)2\pi},0.3e^{(0.3)2\pi},0.3e^{(0.3)2\pi})} = \{\{j_{1},j_{2},j_{4},j_{5}\},\{j_{1},j_{3},j_{4}\},\{j_{2},j_{3},j_{4},j_{5}\}\}.\]  
_Clearly, \(\mathcal{F}_{s}(\mathcal{H})=\{(0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi},0.9e^{(0.9) 2\pi}),(0.6e^{(0.6)2\pi},0.6e^{(0.6)2\pi}),\\ (0.3e^{(0.3)2\pi},0.3e^{(0.3)2\pi},0.3e^{(0.3)2\pi})\}\). Also, \(T_{r}(\mathcal{H})=\{\tau_{1},\tau_{2},\tau_{3}\}=T_{r}^{*}(\mathcal{H})\), where_  
\[\tau_{1} = \{(j_{1},0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi}),(j_{2 },0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi})\},\] \[\tau_{2} = \{(j_{1},0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi}),(j_{3 },0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi})\},\] \[\tau_{3} = \{(j_{2},0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi}),(j_{3 },0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi},0.9e^{(0.9)2\pi})\}.\]  
_Since, \(\{j_{4},j_{5}\}\in T_{r}(\mathcal{H}^{(0.6e^{(0.6)2\pi},0.6e^{(0.6)2\pi},0.6e^{ (0.6)2\pi})})\) and \(\{j_{4}\}\in T_{r}(\mathcal{H}^{(0.3e^{(0.3)2\pi},0.3e^{(0.3)2\pi},0.3e^{(0.3) 2\pi})})\), i.e., no minimal transversal of \(\mathcal{H}^{(0.3e^{(0.3)2\pi},0.3e^{(0.3)2\pi},0.3e^{(0.3)2\pi})}\) contains \(\{j_{4},j_{5}\}\). Thus, \((\mathcal{H}^{(0.6e^{(0.6)2\pi},0.6e^{(0.6)2\pi},0.6e^{(0.6)2\pi})},\mathcal{H }^{(0.3e^{(0.3)2\pi},0.3e^{(0.3)2\pi},0.3e^{(0.3)2\pi})})\) is not \(T\)-related, therefore \(\mathcal{H}\) is not \(T\)-related._  
**Theorem 2**: _Let \(\mathcal{H}=(\mathcal{N},\lambda)\) be an ordered CNHG, then \(T_{r}(\mathcal{H})=T_{r}^{*}(\mathcal{H})\Leftrightarrow\mathcal{H}\) is \(T\)-related._  
**Proof. ** In view of Theorem 1, this is enough to prove that \(T_{r}(\mathcal{H})=T_{r}^{*}(\mathcal{H})\) implies \(\mathcal{H}\) is \(T\)-related. Suppose that \(\mathcal{F}_{s}(\mathcal{H})=\{(\alpha_{1}e^{\Theta_{1}},\beta_{1}e^{\Theta_{1}}, \gamma_{1}e^{\Psi_{1}}),(\alpha_{2}e^{\Theta_{2}},\beta_{2}e^{\Phi_{2}},\gamma_{2}e ^{\Psi_{2}}),\cdots,(\alpha_{n}e^{\Theta_{n}},\beta_{n}e^{\Phi_{n}},\gamma_{n}e^ {\Psi_{n}})\}\) and \(\mathcal{H}\) is not \(T\)-related.