# Complex Neutrosophic Hypergraphs: New Social Network Models
## 2 Complex Neutrosophic Hypergraphs
: _A complex neutrosophic hypergraph (CNHG) on \(\mathcal{J}\) is defined as an ordered pair \(\mathcal{H}=(\mathcal{N},\lambda)\), where \(\mathcal{N}=\{N_{1},N_{2},\cdots,N_{k}\}\) is a finite family of CNSs on \(\mathcal{J}\) and \(\lambda\) is a CNR on CNSs \(N_{j}\)'s such that_  
**(i)**: \[t_{\lambda}(\{r_{1},r_{2},\cdots,r_{l}\}) \leq \min\{t_{N_{j}}(r_{1}),t_{N_{j}}(r_{2}),\cdots,t_{N_{j}}(r_{l})\},\] \[S=\{(x,t_{S}(x),i_{S}(x),f_{S}(x))|x\in\mathcal{J}\},\]0 \  
**(ii)**: \(\lambda^{(\alpha_{k}e^{\Theta_{k}},\beta_{k}e^{\Theta_{k}},\gamma_{lk}e^{\Psi_{k}})} \subset\lambda^{(\alpha_{k+1}e^{\Theta_{k+1}},\beta_{k+1}e^{\Theta_{k+1}},\gamma_ {lk+1}e^{\Psi_{k+1}})},\)  
_is called the fundamental sequence of \(\mathcal{H}=(\mathcal{N},\lambda)\), denoted by \(\mathcal{F}_{s}(\mathcal{H})\). The set of \((\alpha_{i}e^{\Theta_{j}},\beta_{j}e^{\imath\Theta_{j}},\gamma_{ji}e^{\imath \Psi_{j}})\) -level hypergraphs \(\{\mathcal{H}^{(\alpha_{1}e^{\Theta_{1}}}\beta_{1}e^{\Phi_{1}},\gamma_{1}e^{ \Psi_{1}}),\mathcal{H}^{(\alpha_{2}e^{\Theta_{2}},\beta_{2}e^{\Phi_{2}},\gamma_ {2}e^{\Psi_{2}})},\cdots,\mathcal{H}^{(\alpha_{n}e^{\Theta_{n}},\beta_{n}e^{ \omega_{n}},\gamma_{n}e^{\imath\Psi_{n}})}\}\) is called the set of core hypergraphs or the core set of \(\mathcal{H}\), denoted by \(c(\mathcal{H})\)._  
**Example 2**. : _Consider a CNHG \(\mathcal{H}=(\mathcal{N},\lambda)\) on \(\mathcal{J}=\{r_{1},r_{2},r_{3},r_{4},r_{5},r_{6}\}\). The CNR \(\lambda\) is given as, \(\lambda(\{r_{1},r_{2},r_{3}\})=(0.6e^{\iota(0.6)2\pi},0.4e^{\iota(0.4)2\pi},0.3 e^{\iota(0.3)2\pi})\), \(\lambda(\{r_{1},r_{4}\})=(0.8e^{\iota(0.8)2\pi},0.5e^{\iota(0.5)2\pi},0.4e^{ \iota(0.4)2\pi})\), \(\lambda(\{r_{3},r_{4},r_{5}\})=(0.3e^{\iota(0.3)2\pi},0.2e  
**Definition 13**. : _A CNHG \(\mathcal{H}=(\mathcal{N},\Lambda)\) is ordered if \(c(\mathcal{H})\) is ordered, i.e., if \(c(\mathcal{H})=\{\mathcal{H}^{(a_{1}e^{\Theta_{1}},\beta_{1}e^{\Theta_{1}}, \gamma_{1}e^{\Theta_{1}})},\)\(\mathcal{H}^{(a_{2}e^{\Theta_{2}},\beta_{2}e^{\Theta_{2}},\gamma_{2}e^{\Theta_{2}}, \gamma_{2}e^{\Theta_{2}},\gamma_{2}e^{\Theta_{2}})},\)\(\cdots\)\(\mathcal{H}^{(a_{n}e^{\Theta_{n}},\beta_{n}e^{\Theta_{n}},\gamma_{n}e^{\Theta_{n}})}\}\), then \(\{\mathcal{H}^{(a_{1}e^{\Theta_{1}},\beta_{1}e^{\Theta_{1}},\gamma_{1}e^{ \Theta_{1}})}\subset\mathcal{H}^{(a_{2}e^{\Theta_{2}},\beta_{2}e^{\Theta_{2}}, \gamma_{2}e^{\Theta_{2}})}\subset\cdots\)\(\subset\mathcal{H}^{(a_{n}e^{\Theta_{n}},\beta_{n}e^{\Theta_{n}},\gamma_{n}e^{\Theta_{n}})}\}\)._  
_A CNHG \(\mathcal{H}=(\mathcal{N},\Lambda)\) is simply ordered if \(c(\mathcal{H})\) is simply ordered, i.e., if \(e\in E_{j+1}\setminus E_{j}\), then \(e\
ot\subseteq\mathcal{J}_{j}\)._  
**Definition 19**. : _Let \(\mathcal{H}=(\mathcal{N},\lambda)\) be a CNHG. A complex neutrosophic transversal (CNT) \(\tau\) is a CNS of \(\mathcal{J}\) satisfying the condition \(\xi^{h(\xi)}\cap\tau^{h(\xi)}\
eq\emptyset\), for all \(\xi\in\lambda\), where \(h(\xi)\) is the height of \(\xi\)._  
_A minimal complex neutrosophic transversal \(\tau_{1}\) is the CNT of \(\mathcal{H}\) with the property that if \(\tau\subset\tau_{1}\), then \(\tau\) is not a CNT of \(\mathcal{H}\)._  
Let us denote the family of minimal CNTs of \(\mathcal{H}\) by \(T_{r}(\mathcal{H})\). **Definition 20**. : _A CNT \(\tau\) with the property that \(\tau^{(\alpha e^{\Theta_{1}},\beta e^{\omega_{1}},\gamma e^{\pi\gamma})}\in t _{r}(\mathcal{H}^{(\alpha e^{\Theta_{2}},\beta e^{\omega_{2}},\gamma e^{\pi \gamma})})\), for all \(\alpha,\beta,\gamma\in[0,1]\), and \(\Theta,\Phi,\Psi\in[0,2\pi]\) is called the locally minimal CNT of \(\mathcal{H}\). The collection of all locally minimal CNTs of \(\mathcal{H}\) is represented by \(T_{r}^{*}(\mathcal{H})\)._  
_Please note that \(T_{r}^{*}(\mathcal{H})\subseteq T_{r}(\mathcal{H})\), but the converse is not generally true._  
**Definition 21**. : _Let \(N\) be a CNS on \(\mathcal{J}\).