# A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications
## 5 Complete Bipartite Neutrosophic Graphs
Since \(G^{*}\) is a path, we have for every arbitrary node \(w\) in \(G\) that there exists only one node as \(z\) such that \(w\) and \(z\) are connected by means of one path of length \(m\) and this path contains at least one \(uv\). Hence, for the path \(\lambda:z=z_{0},z_{1},z_{2},...z_{m-1},z_{m}=w\). We get \(T_{M}^{m}(zw)=T_{M}(zz_{1})\wedge...\wedge T_{M}(z_{m-1}w)=c_{1}\), \(I_{M}^{m}(zw)=I_{M}(zz_{1})\vee...\lor I_{M}(z_{m-1}w)=c_{2}\), and \(F_{M}^{m}(zw)=F_{M}(zz_{1})\vee...\lor F_{M}(z_{m-1}w)=c_{3}\). Therefore, \(d_{m}(w)=(T_{M}^{m}(zw),I_{M}^{m}(zw),F_{M}^{m}(zw))=(c_{1},c_{2},c_{3})\), and so, \(G\) is a \((m,(c_{1},c_{2},c_{3}))\)-regular neutrosophic graph. Remark 3: _Let \(G=(N,M)\) be a neutrosophic graph on the path graph \(G^{*}=(V,E)\) with \(2m\) nodes. If \(T_{M}\), \(I_{M}\), and \(F_{M}\) are constant functions and \(T_{N}\), \(I_{N}\), and \(F_{N}\) are not, then \(G\) is not \(td_{m}\)-regular._  
6. \(m\)**-Highly Irregular Neutrosophic Graph**  
In this section, we present the notation of \(m\)-HING and the \(m\)-highly totally irregular neutrosophic graph (HTING). We have proven the necessary and sufficient conditions for which the \(m\)-highly irregular and \(m\)-HTINGs are equivalent, and also, we provide \(m\)-highly irregularity on neutrosophic graph \(G\) where \(G^{m}\) is the cycle, path, or star graph. Definition 28: Let \(G=(N,M)\) be a regular connected neutrosophic graph, then:  
1. \(G\) is said to be \(m\)-HING, if each node of \(G\) is adjacent to the other nodes with the different values of the \(d_{m}\)-degree. 2. \(G\) is said to be \(m\)-HTING, if each node of \(G\) is adjacent to the other nodes with the different values of the \(td_{m}\)-degree. Theorem 4.2: _Let \(G=(N,M)\) be a neutrosophic graph. If \(T_{M}\), \(I_{M}\), and \(F_{M}\) are used to denote the three constant functions and \(m\) represents a positive integer number, then \(G\) is \(m\)-HTING if \(G\) is \(m\)-HING._  
Proof: Suppose that \(G\) is \(m\)-HTING. Then, the value of the \(td_{m}\)-degree of all the pairs of adjacent nodes are different. Let two nodes \(v\) and \(u\) be the pair of adjacent nodes with the distinct \(td_{m}\)-degree. Hence, we have \[td_{m}(u)=\Bigg{[}\sum_{u\
eq w\in V}T_{M}^{m}(uw)+T_{N}(u),\sum_{u\
eq w \in V}I_{M}^{m}(uw)+I_{N}(u),\sum_{u\
eq w\in V}F_{M}^{m}(uw)+F_{N}(u)\Bigg{]}\] \[td_{m}(v)=\Bigg{[}\sum_{v\
eq w\in V}T_{M}^{m}(vw)+T_{N}(v),\sum_ {v\
eq w\in V}I_{M}^{m}(vw)+I_{N}(v),\sum_{v\
eq w\in V}F_{M}^{m}(vw)+F_{N}(v) \Bigg{]}.\] \[\text{Since }td_{m}(u)\
eq td_{m}(v),\text{we have,}\sum_{u\
eq w \in V}T_{M}^{m}(uw)+T_{N}(u)\
eq\sum_{v\
eq w\in V}T_{M}^{m}(vw)+T_{N}(v),\sum_ {u\
eq u\in V}I_{M}^{m}(uw)+I_{N}(u)\
eq\sum_{v\
eq w\in V}I_{M}^{m}(vw)+I_{N}( v)\text{ and }\sum_{u\
eq w\in V}F_{M}^{m}(uw)+F_{N}(u)\
eq\sum_{v\
eq w\in V}F_{M}^{m}(vw)+F_{N}(v).\]  
Since \(T_{N}(u)=T_{N}(v)\), \(I_{N}(u)=I_{N}(v)\), and \(F_{N}(u)=F_{N}(v)\), hence we get  
\(\sum_{u\
eq w\in V}T_{M}^{m}(uw)\
eq\sum_{v\
eq w\in V}T_{M}^{m}(vw),\sum_{u \
eq w\in V}I_{M}^{m}(uw)\
eq\sum_{v\
eq w\in V}I_{M}^{m}(vw)\), and \(\sum_{u\
eq w\in V}F_{M}^{m}(uw)\
eq\sum_{v\
eq w\in V}F_{M}^{m}(vw)\). Therefore, we have,  
\(d_{m}(u)=\Bigg{[}\sum_{u\
eq w\in V}T_{M}^{m}(uw),\sum_{u\
eq w \in V}I_{M}^{m}(uw),\sum_{u\
eq w\in V}F_{M}^{m}(uw)\Bigg{]}\
eq\Bigg{[}\sum_{ v\
eq w\in V}T_{M}^{m}(vw),\)  
\(\sum_{v\
eq w\in V}I_{M}^{m}(vw),\sum_{v\
eq w\in V}F_{M}^{m}(vw)\Bigg{]}=d_{m }(v)\). Hence, any pair of adjacent nodes in \(G\) has distinct a \(d_{m}\)-degree. Then, \(G\) is \(m\)-HING. Conversely, \(G\) is an \(m\)-HING. Then, the value of the \(d_{m}\)-degree of each and every pair of adjacent nodes such as \(u\) and \(v\) is distinct. This implies that \(d_{m}(u)\
eq d_{m}(v)\). Since \(T_{N}(u)=T_{N}(v)\), \(I_{N}(u)=I_{N}(v)\), and \(F_{N}(u)=F_{N}(v)\), hence \(td_{m}(u)=d_{m}(u)+T_{N}(u)\
eq id_{m}(u)=d_{m}(u)+I_{N}(u)\
eq fd_{m}(u)=d_{m }(u)+F_{N}(u)\), and so, any two adjacent nodes in \(G\) has the distinct \(td_{m}\)-degree. Therefore, \(G\) is an \(m\)-HTING. Hence the proof.