# A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications
## 5 Complete Bipartite Neutrosophic Graphs  
Definition 27: A neutrosophic graph \(G=(N,M)\) is called a complete bipartite neutrosophic graph if the vertex set \(V\) can be divided into two nonempty sets, such that for every \(v_{1}\), \(v_{2}\in V_{1}\,\text{or}\,V_{2}\) and for every \(u\in V_{1}\,\text{and}\,\,v\in V_{2}\):  
Theorem 5.1: _Let \(G=(N,M)\) be a complete bipartite neutrosophic graph, such that each partition of \(V\) has nodes._  
1. _If for every_ \(v\in V_{1}\backslash V_{2}\) _or_ \(v\in V_{2}\backslash V_{1}\) _,_ \((t_{N}(v),i_{N}(v),\,f_{N}(v))=(c_{1}\,,\,c_{2},\,c_{3})\) _and for the other nodes as_ \(u\)_,_ \(t_{N}(u)\geq c_{1}\) _,_ \(i_{N}(u)\leq c_{2}\,,\,f_{N}(u)\leq c_{3}\,,\,\) _then for all_ \(1\leq m\leq 2k-1\)_, Gis a_ \(d_{m}\) _regular neutrosophic graph._
2. _Let_ \(W=\{\,v_{1}\,,v_{2}\,,\,...,\,v_{k}\}\) _be a set of nodes, such that it contains at least one node of each partition of_ \(V\)_._  
Proof: Suppose that for every \(v_{1}\in V_{1}\backslash V_{2}\), \((t_{N}(v),i_{N}(v),\,f_{N}(v))=(c_{1}\,,\,c_{2},\,c_{3})\). Therefore, in \(V_{1}\), \(t_{N}\) and \(f_{N}\) are constant functions, and since for every \(v_{2}\in V_{1}\backslash V_{2}\), \(t_{N}(v_{1})\leq t_{N}(v_{1})\),\(i_{N}(v_{1})\geq i_{N}(v_{2})\), \(f_{N}(v_{1})\geq f_{N}(v_{21})\) ). Then, by the previous definition, we have that for every \(uv\in E\), we get \((t_{M}(uv),i_{M}(uv),\,f_{M}(uv))=(c_{1}\,,\,c_{2}\,,\,c_{3})\). Hence, for every \(v\in V\) and \(1\leq m\leq 2k-1\), if \(m\) is odd, then \(d_{m}(v)=k(c_{1}\,,\,c_{2}\,,\,c_{3})\) and if \(m\) is even, then \(d_{m}(v)=(k-1)(c_{1}\,,\,c_{2}\,,\,c_{3})\). Therefore, in both cases, we see that \(G\) is a \(d_{m}\)-regular neutrosophic graph. Now, if for every \(v_{1}\in V_{1}\backslash V_{2}\), \((t_{N}(v),i_{N}(v),\,f_{N}(v))=(c_{1}\,,\,c_{2},\,c_{3})\), then the proof is similar to the proof of (i). (ii) The proof is similar to the previous theorem only by the difference, that every \(v\in V\)_and_\(k+1\leq m\leq 2k-1\); if \(m\) is odd, then and if \(m\) is even, then \(d_{m}(v)=(k-1)(c_{1}\,,\,c_{2}\,,\,c_{3})\). Remark 1: _If \(V_{1}=\{\,v_{1}\,,v_{2}\,,\,...,\,v_{n}\}\) and \(V_{1}=\{\,u_{1}\,,u_{2}\,,\,...,\,u_{m}\}\) such that \(n\
eq m\,,\) then the above theorem does not hold._  
Example 5: _Consider the complete bipartite neutrosophic graph \(G=(N,M)\) in Figure 4 shown below._  
_Since \(G\) is a complete bipartite neutrosophic graph, then for every \(uv\in E_{r}(t_{M}(uv),i_{M}(uv),\,f_{M}(uv))=(0.2,0.3,0.6)\). Hence: \(d_{1}(a)=(04,\,0.6,\,1.2)\), \(d_{1}(b)=(04,\,0.6,\,1.2)\), \(d_{1}(c)=(04,\,0.6,\,1.2)\), \(d_{1}(d)=(0.6,\,0.9,\  
\[d_{2}(a) =(04,\,0.6,\,1.2),\,d_{2}(b)=(04,\,0.6,\,1.2),\,d_{2}(c)=(04,\,0.6,\,1.2),\] \[d_{2}(d)=(02,\,\,03,\,0.6),\,\,d_{2}(e)=(02,\,03,\,0.6)\] \[d_{3}(a)=(04,\,0.6,\,1.2),\,d_{3}(b)=(04,\,0.6,\,1.2),\,d_{3}(c)=(0 4,\,0.6,\,1.2),\] \[d_{3}(d)=(06,\,\,09,\,1.8),\,\,d_{3}(e)=(0.6,\,09,\,1.8)\] \[d_{4}(a)=(04,\,0.6,\,1.2),\,d_{4}(b)=(04,\,0.6,\,1.2),\,d_{4}(c)=( 04,\,0.6,\,1.2),\] \[d_{4}(d)=(0,\,\,1,\,1),\,\,d_{4}(e)=(0,\,\,1,\,1)\]  
_Since \(t_{N}\), \(i_{N}\) and \(f_{N}\) are constant functions, we have for all \(m=1,\,2,\,3,\,4,\,G\) does not follow the condition of the \(d_{m}\)-regular neutrosophic graph._  
**Theorem 9**: _Let \(G=(N,M)\) be a regular neutrosophic graph on cycle graph \(G^{*}=(V,E)\) with \(k\geq 3\) nodes. If for \(k-(n-1)\) arcs of \(G\) as \(uv\), \((T_{M}(uv),I_{M}(uv),F_{M}(uv))=(c_{1},c_{2},c_{3})\) and for arcs as \(ab\), \(T_{M}(ab)\geq c_{1}\), \(I_{M}(ab)\geq c_{2}\), and \(F_{M}(ab)\geq c_{3}\), then for any \(n\leq m\leq k-1\), \(G\) is a \(d_{m}\)-regular neutrosophic graph._  
Let \(G=(N,M)\) be a regular neutrosophic graph on cycle graph \(G^{*}\) with \(k\) nodes and for \(k-(n-1)\) arcs of \(G\) as \(uv\), \((T_{M}(uv),I_{M}(uv),F_{M}(uv))=(c_{1},c_{2},c_{3})\).