# Complex Neutrosophic Hypergraphs: New Social Network Models
## 3 \(T\)-Related Complex Neutrosophic Hypergraphs
Since, \(\mathcal{H}\) is ordered, then there exists an ordered sequence \(t_{k}\supseteq t_{k-1}\supset\cdots\supseteq t_{1}\) of crisp minimal transversals of \(\mathcal{H}^{(\alpha_{k}e^{\Theta_{k}},\beta_{k}e^{\Theta_{k}},\gamma_{k}e^{ \alpha_{k}})}\), \(\mathcal{H}^{(\alpha_{k-1}e^{\Theta_{k-1}},\beta_{k-1}e^{\Theta_{k-1}},\gamma_ {k-1}e^{\alpha_{k-1}})}\), \(\cdots\), \(\mathcal{H}^{(\alpha_{1}e^{\Theta_{1}},\beta_{1}e^{\Theta_{1}},\gamma_{1}e^{ \alpha_{1}})}\), respectively. Let \(  
**Remark 1**. :
* _Example 6 shows that there exists some ordered CNHGs that are not T-related._
* _Every simply ordered CNHG_ \(\mathcal{H}=(\mathcal{N},\lambda)\) _satisfies_ \((T_{r}^{*}(\mathcal{H})^{(\alpha\epsilon^{\Theta},\beta\epsilon^{\Theta},\gamma \epsilon^{\Psi})}=T_{r}(\mathcal{H}^{(\alpha\epsilon^{\Theta},\beta\epsilon^{ \Theta},\gamma\epsilon^{\Psi})})\)_, for all_ \(\alpha\in(0,t(h(\mathcal{H}))]\)_,_ \(\beta\in(0,i(h(\mathcal{H}))]\)_,_ \(\gamma\in(0,f(h(\mathcal{H}))]\)_,_ \(\Theta\in(0,\phi(h(\mathcal{H}))]\)_,_ \(\Phi\in(0,\varphi(h(\mathcal{H}))]\)_,_ \(\Psi\in(0,\psi(h(\mathcal{H}))]\)_._  
**Lemma 2**. : _Let \(H=(\mathcal{J},E)\) be a crisp hypergraph and \(j\) be an arbitrary vertex of \(H\). Then \(j\in\mathcal{E}\in T_{r}(H)\Leftrightarrow j\in E_{k}\in E\) such that for any hyperedge \(E_{l}\
eq E_{k}\) of \(H\), \(E_{l}\
subseteq E_{k}\)._  
**Proposition 1**. : _Let \(H_{1}=(\mathcal{J}_{1},E_{1})\) be a crisp partial hypergraph of \(H=(\mathcal{J},E)\) that is obtained by removing those hyperedges of \(H=(\mathcal{J},E)\) that contain any other edges properly. Then,_  
**(i)**: \(T_{r}(H_{1})=T_{r}(H),\)__**(ii)**: \(\cup T_{r}(H)=\mathcal{J}_{1}\)_._  
**Definition 27**. : _Let \(\mathcal{H}=(\mathcal{N},\lambda)\) be a CNHG. The join of \(\mathcal{H}\), denoted by \(J(\mathcal{H})\), is defined as, \(J(\mathcal{H})=\bigcup\limits_{\rho\in\lambda}\rho\), where \(\lambda\) is the CN hyperedge set of \(\mathcal{H}\)._  
_For every \(\alpha\in(0,t(h(\mathcal{H}))]\), \(\beta\in(0,i(h(\mathcal{H}))]\), \(\gamma\in(0,f(h(\mathcal{H}))]\), \(\Theta\in(0,\phi(h(\mathcal{H}))]\), \(\Phi\in(0,\varphi(h(\mathcal{H}))]\), \(\Psi\in(0,\psi(h(\mathcal{H}))]\), the \((\alpha\epsilon^{\Theta},\beta\epsilon^{\Theta},\gamma\epsilon^{\Psi})\)-level cut of \(J(\mathcal{H})\), i.e., \((J(\mathcal{H}))^{(\alpha\epsilon^{\Theta},\beta\epsilon^{\Theta},\gamma \epsilon^{\Psi})}\) is the set of vertices of \((\alpha\epsilon^{\Theta},\beta\epsilon^{\Theta},\gamma\epsilon^{\Psi})\)-level hypergraph of \(\mathcal{H}\), i.e., \((J(\mathcal{H}))^{(\alpha\epsilon^{\Theta},\beta\epsilon^{\Theta},\gamma \epsilon^{\Psi})}=\mathcal{J}(\mathcal{H}^{(\alpha\epsilon^{\Theta},\beta \epsilon^{\Theta},\gamma\epsilon^{\Psi})})\)._  
**Lemma 3**. : _Let \(\mathcal{H}=(\mathcal{N},\lambda)\) be a CNHG and \(\xi\in T_{r}(\mathcal{H})\). If \(j\in supp(\xi)\), then there exists a CN hyperedge \(\rho\) of \(\mathcal{H}\) such that_  
**(i)**: \(\rho(j)=h(\rho)=\xi(j)>0\)_,_
**(ii)**: \(\xi^{h(\rho)}\cap\rho^{h(\rho)}=\{j\}\)_._  
Proof. : Let \(j_{0}\in supp(\xi)\) such that \(\xi\in T_{r}(\mathcal{H})\) and \(\xi(j_{0})=(\alpha_{0}e^{i\phi_{0}},\beta_{0}e^{i\phi_{0}},\gamma_{0}e^{i \phi_{0}})\). Since every \(\xi_{1}\) that is a transversal of \(\mathcal{H}\) contains a transversal \(\xi\) such that \(\xi\subseteq j(\mathcal{H})\). This implies that \(j_{0}\in\mathcal{N}^{(\alpha_{0}e^{i\phi_{0}},\beta_{0}e^{i\phi_{0}},\gamma_{ 0}e^{i\phi_{0}})}=\mathcal{J}(\mathcal{H}^{(\alpha_{0}e^{i\phi_{0}},\beta_{0} e^{i\phi_{0}},\gamma_{0}e^{i\phi_{0}})})\). Therefore, there exists at least one hyperedge \(\rho\) of \(\mathcal{H}\) such that \(\rho(j_{0})\geq(\alpha_{0}e^{i\phi_{0}},\beta_{0}e^{i\phi_{0}},\gamma_{0}e^{i \phi_{0}})\). Let \(\lambda=\{\lambda_{1},\lambda_{2},\cdots,\lambda_{m}\}\) be the set of hyperedges of \(\mathcal{H}\) and \(\rho(j_{0})\geq(\alpha_{0}e^{i\phi_{0}},\beta_{0}e^{i\phi_{0}},\gamma_{0}e^{i \phi_{0}})\).