# A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications
## 3 Regular, \(d_{m}\)-Regular, and \(td_{m}\)-Regular Neutrosophic Graphs
_A co-weak isomorphism_ \(h\) _from a regular neutrosophic graph_ \(G_{1}\) _to another regular neutrosophic graph_ \(G_{2}\) _is a neutrosophic bijective mapping function_ \(h:V_{1}\to V_{2}\)_, which always satisfies the following:_ \((i)\)__\(\forall a_{1}b_{1}\in E_{1}\)__ \((g)\) _h is a homomorphism._ \((h)\)__\(T_{N_{1}}\left(a_{1}b_{1}\right)=\)_T_\({}_{N_{2}}\left(h\left(a_{1}b_{1}\right)\right),\)_I_\({}_{N_{1}}\left(a_{1}b_{1}\right)=\)_I_\({}_{N_{2}}\left(h\left(a_{1}b_{1}\right)\right),\)_F_\({}_{N_{1}}\left(a_{1}b_{1}\right)=\)_F_\({}_{N_{2}}\left(h\left(a_{1}b_{1}\right)\right)\)_._  
**Definition 24**. : _Let \(G=(N,M)\) be a connected neutrosophic graph. The neutrosophic graph \(G\) is called an HING if and only if all the adjacent nodes of \(G\) are always different neutrosophic degrees._  
**Theorem 3**. : _If \(G_{1}\) and \(G_{2}\) are two isomorphic highly-irregular neutrosophic graphs (HING), then their size and order are always equal._  
Proof. : An isomorphism \(h\) is between two HINGs \(G_{1}\) and \(G_{2}\) with the underlying node set \(V_{1}\) and \(V_{2}\), respectively, then:  
* \(\forall a_{1}\in V_{1},a_{1}b_{1}\in E_{1}\)
* \(T_{M_{1}}\left(a_{1}b_{1}\right)=\)_T_\({}_{B_{2}}\left(h\left(a_{1}\right)h\left(a_{2}\right)\right),\)__ \(I_{M_{1}}\left(a_{1}b_{1}\right)=\)_I_\({}_{M_{2}}\left(h\left(a_{1}\right)h\left(a_{2}\right)\right),\)__ \(F_{M_{1}}\left(a_{1}b_{1}\right)=\)_F_\({}_{M_{2}}\left(h\left(a_{1}\right)h\left(a_{2}\right)\right)\)__  
Therefore, we have:  
\[ \begin{split} S(G_{1})&=\left(\sum_{a_{1}b_{1}\in E _{1}}T_{M_{1}}\left(a_{1}b_{1}\right),\sum_{a_{1}b_{1}\in E_{1}}I_{M_{1}}\left( a_{1}b_{1}\right),\sum_{a_{1}b_{1}\in E_{1}}F_{M_{1}}\left(a_{1}b_{1}\right) \right)\\ &=\left(\sum_{a_{1}b_{1}\in V_{1}}T_{M_{2}}\left(h(a_{1}),h(b_{1} \right)\right),\sum_{a_{1}b_{1}\in V_{1}}I_{M_{2}}\left(h(a_{1}),h(b_{1}) \right),\sum_{a_{1}b_{1}\in V_{1}}F_{M_{2}}\left(h(a_{1}),h(b_{1})\right) \right)\\ &=\left(\sum_{a_{2}b_{2}\in E_{2}}T_{M_{2}}\left(a_{2}b_{2} \right),\sum_{a_{2}b_{2}\in E_{2}}I_{M_{2}}\left(a_{2}b_{2}\right),\sum_{a_{2}b _{2}\in E_{2}}F_{M_{2}}\left(a_{2}b_{2}\right)\right)=S(G_{2})\\ &\end{split} \]  
\[ \begin{split} O(G_{1})&=\left(\sum_{a_{1}\in V_{1}}T_ {N_{1}}\left(a_{1}\right),\sum_{a_{1}\in V_{1}}I_{N_{1}}\left(a_{1}\right), \sum_{a_{1}\in V_{1}}F_{N_{1}}\left(a_{1}\right)\right)\\ &=\left(\sum_{a_{1}\in V_{1}}T_{N_{2}}\left(h(a_{1})\right),\sum_{ a_{1}\in V_{1}}I_{N_{2}}\left(h(a_{1})\right),\sum_{a_{1}\in V_{1}}F_{N_{2}}\left(h(a_{1}) \right)\right)\\ &=\left(\sum_{a_{2}\in V_{2}}T_{N_{2}}\left(a_{2}\right),\sum_{a_ {2}\in V_{2}}I_{N_{2}}\left(a_{2}\right),\sum_{a_{2}\in V_{2}}F_{N_{2}}\left(a_ {2}\right)\right)=O(G_{2})\\ \end{split} \]  
**Theorem 4**. : _Let \(G_{1}\) and \(G_{2}\) be two isomorphic HINGs. Then, the degrees of nodes \(a\) and \(h(a)\) of \(G_{1}\) and \(G_{2}\) are the same._Proof. : If \(G_{1}\) and \(G_{2}\) are two isomorphic HINGs, respectively, then \(T_{M_{1}}(a)=T_{M_{1}}\left(h(a_{1})h(b_{1})\right)\), \(I_{M_{1}}(a)=I_{M_{1}}\left(h(a_{1})h(b_{1})\right)\) and \(F_{M_{1}}(a)=F_{M_{1}}\left(h(a_{1})h(b_{1})\right)\) for all \(a_{1},b_{1}\in V_{1}\). Therefore,  
\[d_{T}(a_{1}) =\sum_{a_{1},b_{1}\in V_{1}}T_{M_{1}}(a_{1}b_{1})=\sum_{a_{1},b_{1 }\in V_{1}}T_{M_{2}}\left(h(a_{1})h(b_{1})\right)=d_{T}(h(a_{1}))\] \[d_{I}(a_{1}) =\sum_{a_{1},b_{1}\in V_{1}}I_{M_{1}}(a_{1}b_{1})=\sum_{a_{1},b_{1 }\in V_{1}}I_{M_{2}}\left(h(a_{1})h(b_{1})\right)=d_{I}(h(a_{1}))\] \[d_{F}(a_{1}) =\sum_{a_{1},b_{1}\in V_{1}}F_{M_{1}}(a_{1}b_{1})=\sum_{a_{1},b_{1 }\in V_{1}}F_{M_{2}}\left(h(a_{1})h(b_{1})\right)=d_{F}(h(a_{1}))\]  
for all \(a_{1}\in V_{1}\). This proves that the degrees of nodes \(u\) and \(h(u)\) of \(G_{1}\) and \(G_{2}\) are preserved. **Definition 25**. : _Let \(G=(N,M)\) be a connected neutrosophic graph. \(G\) is defined as: mm_  
1. _A self-weak neutrosophic complementary isomorphic graph if graph_ \(G\) _is weak isomorphic with_ \(G^{c}\)_._
2. _A self-complementary neutrosophic graph if_ \(G^{c}\cong G\)_._  
**Theorem 5**.