# Complex Neutrosophic Hypergraphs: New Social Network Models
## 3 \(T\)-Related Complex Neutrosophic Hypergraphs  
**Definition 24**. : _A CNHG \(\mathcal{H}=(\mathcal{N},\lambda)\) is \(N\)-tempered CNHG of \(H=(\mathcal{J},E)\) if there exists \(H=(\mathcal{J},E)\), a crisp hypergraph, and a CNS \(N\) such that \(\lambda=\{\delta_{e}|e\in E\}\), where_  
\[t_{\delta}(u)e^{\phi_{\delta}(u)}= \begin{cases}\min\{t_{N}(x)e^{\iota\min\{ \phi_{N}(x)\}}|x\in e\},&\text{if}\quad u\in e,\\ 0,&\text{otherwise}.\end{cases} \]  
\[i_{\delta}(u)e^{\iota\phi_{\delta}(u)}= \begin{cases}\min\{i_{N}(x)e^{\iota\min \{\phi_{N}(x)\}}|x\in e\},&\text{if}\quad u\in e,\\ 0,&\text{otherwise}.\end{cases} \]  
\[f_{\delta}(u)e^{\iota\psi_{\delta}(u)}= \begin{cases}\max\{f_{N}(x)e^{\iota\max \{\psi_{N}(x)\}}|x\in e\},&\text{if}\quad u\in e,\\ 0,&\text{otherwise}\end{cases} \]  
_An N-tempered CNHG \(\mathcal{H}=(\mathcal{N},\lambda)\) determined by \(H\) and CNS \(N\) is denoted by \(N\otimes H\)._  
**Definition 25**. : _A pair \((G,I)\) of crisp hypergraphs is \(T\)-related if whenever \(g\) is a minimal transversal of \(G\), \(k\) is any transversal of \(J\), and \(g\subseteq k\), then there exists a minimal transversal \(t\) of \(J\) such that \(g\subseteq t\subseteq k\)._  
**Definition 26**. : _Let \(\mathcal{H}=(\mathcal{N},\lambda)\) be a CNHG with \(\mathcal{F}_{s}(\mathcal{H})=\{(\alpha_{1}e^{\Theta_{1}},\beta_{1}e^{\iota \Phi_{1}},\gamma_{1}e^{\iota\Psi_{1}}),(\alpha_{2}e^{\Theta_{2}},\beta_{2}e^{ \iota\Phi_{2}},\gamma_{2}e^{\iota\Psi_{2}}),\cdots,(\alpha_{n}e^{\Theta_{n}}, \beta_{n}e^{\Phi_{n}},\gamma_{n}e^{\iota\Psi_{n}})\}\). Then, \(\mathcal{H}\) is \(T\)-related if from the core set_  
\[c(\mathcal{H})=\{\mathcal{H}^{(\alpha_{1}e^{\Theta_{1}},\beta_{1}e^{\Theta_{ 1}},\gamma_{1}e^{\iota\Psi_{1}})},\mathcal{H}^{(\alpha_{2}e^{\Theta_{2}},\beta_ {2}e^{\Theta_{2}},\gamma_{2}e^{\Psi_{2}})},\cdots,\mathcal{H}^{(\alpha_{n}e^{ \Theta_{n}},\beta_{n}e^{\phi_{n}},\gamma_{n}e^{\iota\Psi_{n}})}\}\]  
_of \(\mathcal{H}\), every successive ordered pair \((\mathcal{H}^{(\alpha_{j}e^{\Theta_{j}},\beta_{j}e^{\iota\Phi_{j}},\gamma_{j}e ^{\iota\Psi_{j}})},\mathcal{H}^{(\alpha_{j-1}e^{\Theta_{j-1}},\beta_{j-1}e^{ \phi_{j-1}},\gamma_{j-1}e^{\iota\Psi_{j-1}})})\) is \(T\)-related._  
_If \(\mathcal{F}_{s}(\mathcal{H})\) contains only one element, \(\mathcal{H}\) is considered to be trivially \(T\)-related._  
**Theorem 1**. : _Let \(\mathcal{H}=(\mathcal{N},\lambda)\) be a \(T\)-related CNHG, then \(T_{r}(\mathcal{H})=T_{r}^{*}(\mathcal{H})\)._  
Proof. : Let \(\mathcal{H}=(\mathcal{N},\lambda)\) be a \(T\)-related CNHG with \(\mathcal{F}_{s}(\mathcal{H})=\{(\alpha_{1}e^{\Theta_{1}},\beta_{1}e^{\Phi_{1}}, \gamma_{1}e^{\iota\Psi_{1}}),(\alpha_{2}e^{\Theta_{2}},\beta_{2}e^{\Phi_{2}}, \gamma_{2}e^{\iota\Psi_{2}}),\cdots,(\alpha_{1}e^{\iota\Theta_{n}},\beta_{n}e^{ \epsilon\Phi_{n}},\gamma_{n}e^{\iota\Psi_{n}})\}\). Then, there arises two cases:  
**Case (i)**: First we consider that \(\mathcal{F}_{s}(\mathcal{H})=\{(\alpha_{1}e^{\Theta_{1}},\beta_{1}e^{\Phi_{1}}, \gamma_{1}e^{\iota\Psi_{1}})\}\). Then, Lemma 1 implies that for each \(\xi\in T_{r}(\mathcal{H})\), \(\xi^{(\alpha_{2}e^{\Theta_{2}},\beta_{2}e^{\Phi_{2}},\gamma_{2}e^{\iota\Psi})}\in T _{r}(\mathcal{H}^{(\alpha_{2}e^{\Theta_{2}},\beta_{2}e^{\Phi_{2}},\gamma_{2}e^{ \iota\Psi})})\), for all \(0<\alpha\leq t(h(\mathcal{H}))\), \(0<\beta\leq i(h(\mathcal{H}))\), \(\gamma\geq f(h(\mathcal{H}))>0\), \(0<\Theta\leq\phi(h(\mathcal{H}))\), \(0<\Phi\leq\phi(h(\mathcal{H}))\), and \(\Psi\geq\psi(h(\mathcal{H}))>0\). Thus, \(T_{r}(\mathcal{H})=T_{r}^{*}(\mathcal{H})\). **Case (ii)**: We now suppose that \(|\mathcal{F}_{s}(\mathcal{H})|\geq 2\). Since, \(T_{r}^{*}(\mathcal{H})\subseteq T_{r}(\mathcal{H})\), we just have to prove that \(T_{r}(\mathcal{H})\subseteq T_{r}^{*}(\mathcal{H})\). Let \(\xi\in T_{r}(\mathcal{H})\), and \(\xi^{(\alpha_{1}e^{\Theta_{1}},\beta_{1}e^{\Phi_{1}},\gamma_{1}e^{\Psi_{1}})} \subset\xi^{(\alpha_{2}e^{\Theta_{2}},\beta_{2}e^{\Phi_{2}},\gamma_{2}e^{\Psi_{2}})}\).