# A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications
## 5 Complete Bipartite Neutrosophic Graphs
Theorem 4.1: _Let \(G=(N,M)\) be a neutrosophic graph on cycle graph \(G^{*}=(V,E)\) with \(k\geq 3\) nodes and for all \(i=1,2,...,k-1\) (where \(v_{k+1}=v_{1}\)), \(T_{M}(v_{i}v_{i+1})<T_{M}(v_{i+1}v_{i+2})or\)\(I_{M}(v_{i}v_{i+1})>I_{M}(v_{i+1}v_{i+2})\) or \(F_{M}(v_{i}v_{i+1})>F_{M}(v_{i+1}v_{i+2})\). Then, \(G\) is a \(\Bigg{[}1,2,3,\left[\frac{k}{2}\right]\Bigg{]}\)-HING._  
Proof. : Suppose that \(v_{1},v_{2},...,v_{k},v_{1}\) is the arcs of \(G\), where \(T_{M}(v_{1}v_{2})<T_{M}(v_{2}v_{3})<...<T_{M}(v_{i}v_{i+1})<T_{M}(v_{i+1}v_{i+2 })<...<T_{M}(v_{k}v_{1})\) and there exists \(v_{i}v_{i+1}\in E\) such that for \(1\leq m\leq\left[\frac{k}{2}\right]\), \(d_{m}(v_{i})=d_{m}(v_{i+1})\). Therefore, we get \(\sum_{v_{i}\
eq v_{i+1}\in V}T_{M}^{m}(v_{i}v_{j})=T_{M}(v_{i}v_{i+1})+T_{M}(v_{ j}v_{j+1})\) and \(\sum_{v_{i+1}\
eq v_{j}\in V}T_{M}^{m}(v_{i}v_{j})=T_{M}(v_{i}v_{i+1})+T_{M}(v_{ j+1}v_{j+2})\). Since \(d_{m}(v_{i})=d_{m}(v_{i+1})\), \(T_{M}(v_{i}v_{i+1})+T_{M}(v_{j}v_{j+1})=T_{M}(v_{i}v_{i+1})+T_{M}(v_{j+1}v_{j+2})\), thus \(T_{M}(v_{i}vi+1)=T_{M}(v_{j+1}vj+2)\), which is a contradiction. Now, for every \(v_{1},v_{2},...,v_{k},v_{1}\in E\), also we have  
\(I_{M}(v_{1}v_{2})>I_{M}(v_{2}v_{3})>...>I_{M}(v_{i}v_{i+1})>I_{M}(v_{i+1}v_{i+2 })>...>I_{M}(v_{k}v_{1})\),  
\(F_{M}(v_{1}v_{2})>F_{M}(v_{2}v_{3})>...>F_{M}(v_{i}v_{i+1})>F_{M}(v_{i+1}v_{i+2 })>...>F_{M}(v_{k}v_{1})\). The proof is similar to the case as true membership, and the contradiction is obtained. Therefore, \(G\) is an \(\Bigg{[}1,2,3,\left[\frac{k}{2}\right]\Bigg{]}\)-HING. Remark 4: _Let \(G=(N,M)\) be a neutrosophic graph on cycle graph \(G^{*}=(V,E)\) with \(k\geq 3\) nodes and for all \(i=1,2,...,k-1\) (where \(v_{k+1}=v_{1}\)), \(T_{M}(v_{i}v_{i+1})<T_{M}(v_{i+1}v_{i+2})\)or \(I_{M}(v_{i}v_{i+1})>I_{M}(v_{i+1}v_{i+2})\) or \(F_{M}(v_{i}v_{i+1})>F_{M}(v_{i+1}v_{i+2})\). Then, \(G\) is not a \(\Bigg{[}1,2,3,\left[\frac{k}{2}\right]\Bigg{]}\)-HING in general._  
Example 6: _Consider the neutrosophic graph \(G=(N,M)\) on graph \(G^{*}=(V,E)\) shown in the \[d_{2}(d)=(02,\,\,03,\,0.6),\,\,d_{2}(e)=(02,\,03,\,0.6)\]0_We see that \(T_{M}(ab)<T_{M}(bc)<T_{B}(cd)<T_{B}(da)\), and we get \(td_{2}(a)=(0.3+0.4,0.6+0.4)=(0.7,1.0)\)\(td_{2}(b)=(0.2+0.3,0.6+0.6)=(0.5,1.2)\)\(td_{2}(c)=(0.3+0.4,0.6+0.5)=(0.7,1.1)\)\(td_{2}(d)=(0.2+0.5,0.6+0.4)=(0.7,1.0)\)  
Since for two vertices \(a\) and \(b\), they are adjacent, \(td_{2}(a)=td_{2}(b)\). Then, \(G\) is not a two-HTING. Let \(G=(N,M)\) be a neutrosophic graph on cycle graph \(G^{*}=(V,E)\) with \(k\geq 3\) nodes and for all \(i=1,2,...,k-1\) (where \(v_{k+1}=v_{1}\)), \(T_{M}(v_{i}v_{i+1})<T_{M}(v_{i+1}v_{i+2})\)or \(I_{M}(v_{i}v_{i+1})>I_{M}(v_{i+1}v_{i+2})\) or \(F_{M}(v_{i}v_{i+1})>F_{M}(v_{i+1}v_{i+2})\). Then, \(G\) is a \(\left[1,2,3,\left[\frac{k}{2}\right]\right]\)-HING. Proof. : Suppose that \(v_{1},v_{2},...,v_{k},v_{1}\) is the arcs of \(G\), where \(T_{M}(v_{1}v_{2})<T_{M}(v_{2}v_{3})<...<T_{M}(v_{i}v_{i+1})<T_{M}(v_{i+1}v_{i+2 })<...<T_{M}(v_{k-1}v_{k})\). Then, for every \(1\leq m\leq\left[\frac{k}{2}\right]\), we get  
\[\sum_{v_{i}\
eq v_{j}\in V}T_{M}^{m}(v_{i}v_{j})=\left\{ \begin{array}{l}T_{ M}(v_{i}v_{i+1}),\;if\;i=1,2,3,...,m\\ T_{M}(v_{i}v_{i+1})+T_{M}(v_{i-m}v_{i-m+11},\;if\;i=m+1,...,k-m\\ T_{M}(v_{i-m}v_{i-m+1},\;if\;i=k-m+1,...k.\end{array} \right.\]  
Hence, for every two adjacent nodes as \(v_{i}\) and \(v_{i+1}\) in \(G\), we get \(d_{m}(v_{i})\
eq d_{m}(v_{i+1})\). Therefore, \(G\) is a \(\left[1,2,3,\left[\frac{k}{2}-1\right]\right]\)-HING. Now, suppose that \(m=\left[\frac{k}{2}\right]\). If \(k\) is even, then we have  
\[\sum_{v_{i}\
eq v_{j}\in V}T_{M}^{m}(v_{i}v_{j})=\left\{ \begin{array}{l}T_{ M}(v_{i}v_{i+1}),\;if\;i=1,2,3,...,m\\ T_{M}(v_{i-m}v),\;if\;i=m+1,...,k-m\end{array} \right.\]  
If \(k\) is odd, then we have  
\[\sum_{v_{i}\
eq v_{j}\in V}T_{M}^{m}(v_{i}v_{j})=\left\{ \begin{array}{l}T_{ M}(v_{i}v_{i+1}),\;if\;i=1,2,3,...,m\\ T_{M}(v_{1}v_{2})+T_{M}(v_{i-m}v_{i-m+11},\;if\;i=m+1\\ T_{M}(v_{i-m}v_{i-m+1},\;if\;i=m+2,...k.\end{array}