# A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications
## 3 Regular, \(d_{m}\)-Regular, and \(td_{m}\)-Regular Neutrosophic Graphs
\begin{array}{l}d_{2}(a)=\left(\left(\left(0.2+0.2\right)+0.3\right),\left(\left(0.6 +0.8\right)+0.6\right),\left(\left(0.8+0.9\right)+0.7\right)\right)=\left(0.7,2. 0,2.4\right)\\ td_{2}(b)=\left(\left(\left(0.2+0.2\right)+0.3\right),\left(\left(0.8+0.8 \right)+0.6\right),\left(\left(0.9+0.9\right)+0.6\right)\right)=\left(0.7,2. 1,2.4\right)\\ td_{2}(a)=\left(\left(\left(0.2+0.2\right)+0.4\right),\left(\left(0.8+0.8 \right)+0.5\right),\left(\left(0.9+0.9\right)+0.6\right)\right)=\left(0.8,2. 1,2.4\right)\end{array} \]  
**Definition 19**. : _Let \(G=\left(N,M\right)\) is a neutrosophic graph. \(G\) is an \(\left(m,\left(k_{1},k_{2},k_{3}\right)\right)\)-totally regular neutrosophic graph or \(td_{m}\)-regular neutrosophic graph if for all nodes \(a\in V\) in \(G\), \(td_{m}(a)=\left(k_{1},k_{2},k_{3}\right)\)._  
**Example 4**. : _An example of a \(\left(m,\left(k_{1},k_{2},k_{3}\right)\right)\)-totally regular neutrosophic graph is shown in Figure 3. It is a (2, (0.6, 2.0,2.1))-regular neutrosophic graph._  
**Theorem 1**. : _Let \(G=\left(N,M\right)\) be an SVNG. If \(T_{N}\), \(I_{N}\)and \(F_{N}\) are constant functions, then \(G\) is a \(d_{m}\)-regular neutrosophic graph if \(G\) is a \(td_{m}\)-regular neutrosophic graph (\(m\) is a positive integer)._  
Proof. : Suppose that for every node \(v\) in \(G,\left(T_{N}\left(v\right),I_{N}\left(v\right),F_{N}\left(v\right)\right)= \left(c_{1},c_{2},c_{3}\right)\) and \(d_{m}\left(v\right)=\left(d_{1},d_{2},d_{3}\right)\). Then:  
\[td_{m}\left(v\right)=d_{m}\left(v\right)+\left(T_{N}\left(v\right),I_{N}\left( v\right),F_{N}\left(v\right)\right)=\left(d_{1}+c_{1},d_{2}+c_{2},d_{3}+c_{3} \right);\]  
hence, \(G\) is a \(td_{m}\)-regular neutrosophic graph. If \(G\) is a \(td_{m}\)-regular neutrosophic graph, then the proof is similar to the previous case. **Theorem 2**. : _Let \(G=\left(N,M\right)\) be an \(\left(m,\left(d_{1},d_{2},d_{3}\right)\right)\)-totally regular and an \(\left(m,\left(k_{1},k_{2},k_{3}\right)\right)\)-totally regular neutrosophic graph with \(n\) nodes. Then, \(T_{N},I_{N}\) and \(F_{N}\) are constant functions, and \(O(G)=n(k_{1}-d_{1},k_{2}-d_{2},k_{3}-d_{3})\)._  
Proof. : If \(G\) is an \(\left(m,\left(d_{1},d_{2},d_{3}\right)\right)\)-regular neutrosophic graph and an \(\left(m,\left(k_{1},k_{2},k_{3}\right)\right)\)-totally regular neutrosophic graph respectively, then for all \(v\in V\) we get,  
\[d_{m}(u)=\left(d_{1},d_{2},d_{3}\right)\rightarrow\left(\sum_{u\
eq\upsilon\in V }T_{M}^{m}(uv),\sum_{u\
eq\upsilon\in V}I_{b}^{m}(uv),\sum_{u\
eq\upsilon\in V }F_{b}^{m}(uv)\right)=\left(d_{1},d_{2},d_{3}\right)\]  
and:  
\[td_{m}(v)=\left(k_{1},k_{2},k_{3}\right)\rightarrow\left(\sum_{u\
eq\upsilon \in V}T_{B}^{m}(uv),+T_{N}(v),\sum_{u\
eq\upsilon\in V}I_{M}^{m}(uv),+I_{N}(v), \sum_{u\
eq\upsilon\in V}F_{M}^{m}(uv),+F_{N}(v)\right)=\left(k_{1},k_{2},k_{3}\right).\]  
Figure 3: A neutrosophic graph \(G\). Hence, by (1), \(\left(d_{1},d_{2},d_{3}\right)=\left(k_{1}-T_{N}\left(v\right),k_{2}-I_{N}\left(v \right),k_{3}-F_{N}\left(v\right)\right)\) and so \(\left(T_{N}\left(v\right),I_{N}\left(v\right),F_{N}\left(v\right)\right)= \left(k_{1}-d_{1},k_{2}-d_{2},k_{3}-d_{3}\right)\). Then, \(T_{N}J_{N}\) and \(F_{N}\) are constant functions, and since \(G\) has \(n\) nodes, we get:  
\[O(G)=\left(\sum_{v\in V}T_{N}(v),\sum_{v\in V}I_{N}(v),\sum_{v\in V}F_{N}(v) \right)=n(k_{1}-d_{1},k_{2}-d_{2},k_{3}-d_{3}).\]  
_Free and Busy Vertex on the Regular Neutrosophic Graph_  
**Definition 20**. : _Let \(G=\left(N,M\right)\) be a regular neutrosophic graph and \(a\in V\) be a node of \(G\).