# A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications
## 5 Complete Bipartite Neutrosophic Graphs
Since \(G^{*}\) is a cycle, we have for every node \(w\) in \(G\) that there exist exactly two distinct nodes \(w_{1}\) and \(w_{2}\) such that \(w\) and \(w_{1}\) are connected by means of one path of length \(m\) (where \(n\leq m\leq k-1\)) and each path contains at least one arc \(uv\). Hence, for two paths,  
\(\lambda_{1}:w_{1}=u_{0},u_{1},u_{2},...u_{m-1},u_{m}=w\) and \(\lambda_{2}:w_{1}=v_{0},v_{1},v_{2},...v_{m-1},v_{m}=w\). Therefore, we obtain  
\(T_{M}^{m}(w_{1}w)+T_{M}^{m}(w_{2}w)=T_{M}(w_{1}u_{1})\wedge...\wedge T_{M}(u _{m-1}w)+T_{M}(w_{2}v_{1})\wedge...\wedge T_{M}(v_{m-1}w)=c_{1}+c_{1}=2c_{1}\). \(I_{M}^{m}(w_{1}w)+I_{M}^{m}(w_{2}w)=I_{M}(w_{1}u_{1})\vee...\lor I_{M}(u_{m-1 }w)+I_{M}(w_{2}v_{1})\vee...\lor I_{M}(v_{m-1}w)=c_{1}+c_{1}=2c_{1}\). \(\square\)  
**Remark 2**: _Let \(G\) be a regular neutrosophic graph on cycle graph \(G^{*}\) with \(k\) nodes. If \(T_{M}\), \(I_{M}\), and \(F_{M}\) are constant functions and \(T_{N}\), \(I_{N}\), and \(F_{N}\) are not, then for any \(1\leq m\leq k-1\), \(G\) is not a \(td_{m}\)-regular neutrosophic graph in general._  
**Theorem 10**: _Let \(G=(N,M)\) be a neutrosophic graph on cycle graph \(G^{*}=(V,E)\) with \(k\geq 3\) nodes. If \(G\) is a \(d_{1}\)-regular neutrosophic graph, then \(T_{M}\), \(I_{M}\), and \(F_{M}\) are constant functions._  
Suppose that \(G\) is a \(d_{1}\)-regular neutrosophic graph and there exists \(uv\in E\) such that \((T_{M}(uv),I_{M}(uv),F_{M}(uv))=(c_{1},c_{2},c_{3})\), where for every \(ab\in E-\{uv\}\), \((T_{M}(ab),I_{M}(ab),F_{M}(ab))=(c_{1},c_{2},c_{3})\). Then, \(T_{M}(uv)\
eq c_{1}\), \(I_{M}(uv)\
eq c_{2}\), and \(F_{M}(uv)\
eq c_{3}\). Case \(1:\) If \(T_{M}(uv)\
eq c_{1}\),  
\(\sum\limits_{u\
eq z\in V}T_{M}^{1}(uz)=c_{1}+T_{M}(uv)\), \(\sum\limits_{v\
eq z\in V}T_{M}^{1}(vz)=c_{1}+T_{M}(uv)\). Since \(G\) has \(k\geq 3\) nodes, there exists \(w\in V\) such that \(w\
eq v,u\). Hence, we get \(\sum\limits_{u\
eq z\in V}T_{M}^{1}(wz)=2c_{1}\), which implies that \(T_{M}^{1}(wz)\
eq T_{M}^{1}(vz)\) and \(T_{M}^{1}(wz)\
eq T_{M}^{1}(uz)\). Therefore, \(d_{1}(w)\
eq d_{1}(u)\) and \(d_{1}(w)\
eq d_{1}(v)\); hence, \(G\) is not a \(d_{1}\)-regular neutrosophic graph, which is a contradiction. Case 2 : If \(I_{M}(uv)\
eq c_{2}\),  
\[\sum_{u\
eq z\in V}I_{M}^{1}(uz)=c_{2}+I_{M}(uv),\ \sum_{v\
eq z\in V}I_{M}^{1}( vz)=c_{2}+I_{M}(uv).\]  
Since \(G\) has \(k\geq 3\) nodes, there exists \(w\in V\) such that \(w\
eq v,u\). Hence, we get \(\sum_{u\
eq z\in V}I_{M}^{1}(wz)=2c_{2}\), which implies that \(I_{M}^{1}(wz)\
eq I_{M}^{1}(vz)\) and \(I_{M}^{1}(wz)\
eq I_{M}^{1}(uz)\). Therefore, \(d_{1}(w)\
eq d_{1}(u)\) and \(d_{1}(w)\
eq d_{1}(v)\); hence, \(G\) is not a \(d_{1}\)-regular neutrosophic graph, which is a contradiction. Case 3 : If \(F_{M}(uv)\
eq c_{3}\),  
\(\sum_{u\
eq z\in V}F_{M}^{1}(uz)=c_{3}+F_{M}(uv)\), \(\sum_{v\
eq z\in V}F_{M}^{1}(vz)=c_{3}+F_{M}(uv)\). Since \(G\) has \(k\geq 3\) nodes, there exists \(w\in V\) such that \(w\
eq v,u\). Hence, we get \(\sum_{u\
eq z\in V}F_{M}^{1}(wz)\ =\ 2c_{3}\), which implies that \(F_{M}^{1}(wz)\
eq F_{M}^{1}(vz)\) and \(F_{M}^{1}(wz)\
eq F_{M}^{1}(uz)\). Therefore, \(d_{1}(w)\
eq d_{1}(u)\) and \(d_{1}(w)\
eq d_{1}(v)\); hence, \(G\) is not a \(d_{1}\)-regular neutrosophic graph, which is a contradiction. Hence the proof. Theorem 4.1: _Let \(G=(N,M)\) be a neutrosophic graph on the path graph \(G^{*}=(V,E)\) with \(2m\) nodes. If for at least \(m+1\) arcs of \(G\) as \(uv\), \((T_{M}(uv),I_{M}(uv),F_{M}(uv))=(c_{1},c_{2},c_{3})\) and for other arcs as \(ab\), \(T_{M}\geq c_{1}\), \(I_{M}\leq c_{2}\) and \(F_{M}\leq c_{3}\), then \(G\) is also a \(d_{m}\)-regular neutrosophic graph._  
Proof: Suppose that \(G^{*}\) is a path with \(2m\) nodes and for at least \(m+1\)arcs of \(G\) as \(uv\), \((T_{M}(uv),I_{M}(uv),F_{M}(uv))=(c_{1},c_{2},c_{3})\).