# A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications
## 3 Regular, \(d_{m}\)-Regular, and \(td_{m}\)-Regular Neutrosophic Graphs  
In this section, first we define the regular neutrosophic graph, regular strong neutrosophic graph, \(d_{m}\)-degree, and \(td_{m}\)-degree of nodes in a neutrosophic graph. Then, we propose the notions of \(d_{m}\) and \(td_{m}\)-regular neutrosophic graphs and prove the necessary and sufficient conditions, for which under these conditions, \(d_{m}\)-regular with \(td_{m}\)-regular neutrosophic graphs are equivalent. Definition 14: Let \(G=\left(N,M\right)\) be an SVNG. \(G\) is a regular neutrosophic graph if it satisfies the following conditions. \[\sum_{a\
eq b}T_{M}(a,b)=\text{Constant},\sum_{a\
eq b}I_{M}(a,b)=\text{ Constant},\sum_{a\
eq b}F_{M}(a,b)=\text{Constant}\]  
Definition 15: Let \(G=\left(N,M\right)\) be an SVNG. \(G\) is a regular strong neutrosophic graph if it satisfies the following conditions. \[ \begin{array}{l}T_{M}(a,b)=\min\left(T_{N}\left(a\right),T_{N}\left(b \right)\right)\text{ \ and \ }\sum_{a\
eq b}T_{M}(a,b)=\text{Constant}\\ I_{M}(a,b)=\max\left(I_{N}\left(a\right),I_{N}\left(b\right)\right)\text{ \ and \ }\sum_{a\
eq b}I_{M}(a,b)=\text{Constant}\\ F_{M}(a,b)=\max\left(F_{N}\left(a\right),F_{N}\left(b\right)\right)\text{ \ and \ }\sum_{a\
eq b}F_{M}(a,b)=\text{Constant}\end{array} \]  
Definition 16: Let \(G=\left(N,M\right)\) be an SVNG and the \(d_{m}\)-degree of any node \(a\) in \(G\) be denoted as \(d_{m}(a)\) where:  
\[d_{m}\left(a\right)=\left(\sum_{a\
eq b\in V}T_{M}^{m}\left(a,b\right),\sum_{ a\
eq b\in V}I_{M}^{m}\left(a,b\right),\sum_{a\
eq b\in V}F_{M}^{m}\left(a,b \right)\right)\]  
Here, the path \(a,a_{1},a_{2},...,a_{m-1},b\) is the shortest path between the nodes \(u\) and \(v\), and the length of this path is \(m\). Example 1: We have considered an example of an SVNG \(G\), presented in Figure 1.  
_Then, the \(d_{2}\)-degree of the nodes in \(G\) is computed as follows._  
\[ \begin{array}{l}d_{2}(a)=(0.2+0.2,0.6+0.8,0.8+0.9)=(0.4,1.4,1.7)\\ d_{2}(b)=(0.2+0.2,0.8+0.8,0.9+0.9)=(0.4,1.6,1.8)\\ d_{2}(c)=(0.2+0.2,0.6+0.8,0.8+0.9)=(0.4,1.6,1.8)\end{array} \]  
**Definition 17**. : _Let \(G=(N,M)\) be an SVNG and \(a\in V\) be a node \(G\). The total \(d_{m}\)-degree (\(td_{m}\)-degree) of node \(a\) in \(G\) is computed as follows._  
\[td_{m}\left(a\right)=\left(\left(\sum_{a\
eq b\in V}T_{M}^{m}\left(a,b\right)+ T_{N}\left(a\right)\right),\left(\sum_{a\
eq b\in V}I_{M}^{m}\left(a,b\right)+ I_{N}\left(a\right)\right),\left(\sum_{a\
eq b\in V}F_{M}^{m}\left(a,b\right)+ F_{N}\left(a\right)\right)\right)\]  
_Here, the path \(a,a_{1},a_{2},...,a_{m-1},b\) is the shortest path between the nodes \(u\) and \(v\), and the length of this path is \(m\)._  
**Example 2**. : _Let us consider an example of an SVNG, shown in Figure 1. Then, the \(td_{2}\)-degree of the nodes in \(G\) is as follows._  
\[ \begin{array}{l}td_{2}(a)=\left(\left(\left(0.2+0.2\right)+0.3\right),\left( \left(0.6+0.8\right)+0.6\right),\left(\left(0.8+0.9\right)+0.7\right)\right)= \left(0.7,2.0,2.4\right)\\ td_{2}(b)=\left(\left(\left(0.2+0.2\right)+0.3\right),\left(\left(0.8+0.8 \right)+0.6\right),\left(\left(0.9+0.9\right)+0.6\right)\right)=\left(0.7,2.1, 2.4\right)\\ td_{2}(a)=\left(\left(\left(0.2+0.2\right)+0.4\right),\left(\left(0.8+0.8 \right)+0.5\right),\left(\left(0.9+0.9\right)+0.6\right)\right)=\left(0.8,2.1, 2.4\right)\end{array} \]  
**Definition 18**. : _Let \(G=(N,M)\) be a neutrosophic graph. \(G\) is said to be a \((m,(d_{1},d_{2},d_{3}))\)-regular neutrosophic graph or \(d_{m}\)-regular if for all nodes \(a\in V\) in \(G\), \(d_{m}(a)=(d_{1},d_{2},d_{3})\)._  
**Example 3**. : _An example of an \((m,(d_{1},d_{2},d_{3}))\)-totally regular neutrosophic graph is pictured in Figure 2. Figure 2 is a (2, (0.1, 0.6,0.9))-regular neutrosophic graph._  
Figure 1: A neutrosophic graph \(G\). Figure 2: A neutrosophic graph \(G\). \[