# A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications
## 2 Preliminaries  
In this portion, we will discuss the single-valued neutrosophic graph (SVNG), adjacent node, path, isolated node, strength of a path, strong SVNG, complement SVNG, and complete SVNG, which are efficient for the present work. Definition 1: Let \(G=(N,M)\) be an SVNG [42; 43] where \(N\) and \(M\) are represented by two neutrosophic sets on \(V\) and \(E\), respectively, which satisfy the following. \[ \begin{array}{l}T_{M}(a,b)\leq\min\left(T_{N}\left(a\right),T_{N}\left(b \right)\right)\\ I_{M}(a,b)\geq\max\left(I_{N}\left(a\right),I_{N}\left(b\right)\right)\\ F_{M}(a,b)\geq\max\left(F_{N}\left(a\right),F_{N}\left(b\right)\right)\end{array} \]  
Here, \(a\) and \(b\) are two vertices of \(G,and\)\(\left(a,b\right)\in E\) is an edge of \(G\). Definition 2: The two nodes \(a\) and \(b\) of an SVNG \(G=(N,M)\) are defined as the adjacent nodes in \(G\) if and only if \(\left(T_{M}(a,b),I_{M}(a,b),F_{M}(a,b)\right)>0\). The two nodes \(a\) and \(b\) are called the neighbor nodes, and \(\left(a,b\right)\) is incident at \(a\) and \(b\). Definition 3: Let \(G=(N,M)\) be an SVNG and \(P\) be a path of \(G\). \(P\) is a collection of different nodes, \(a_{0},a_{1},a_{2},...,a_{n}\) such that \(\left(T_{M}\left(a_{i-1},a_{i}\right),I_{M}\left(a_{i-1},a_{i}\right),F_{M} \left(a_{i-1},a_{i}\right)\right)>0\) for \(0\leq i\leq n\). Here, \(n\) represents the neutrosophic length of the path \(P\). A single neutrosophic vertex, i.e., \(a_{i}\) in \(G\), is also assumed as a path. The path length of a single node \(a_{i}\) is \(\left(0,0,0\right)\). We define the order pair \(\left(a_{i-1},a_{i}\right)\) as the edge of the path. \(P\) is said to be a neutrosophic cycle if \(a_{0}=a_{n}\) and \(n\geq 3\). Definition 4: A node \(a_{i}\in V\) of an SVNG \(G\) is called the isolated node if there exists no incident arc to the vertex \(a_{i}\). **Definition 5**. : _Let \(G=(N,M)\) be an SVNG. If \(G\) has a path \(P\) of path length [42]\(k\) from node \(a\) to node \(b\) in \(G\) such as \(P=\left\{a=a_{1},\left(a_{1},a_{2}\right),a_{2},...,a_{k-1}\left(a_{k-1},a_{k} \right),a_{k}=b\right\}\), then \(T_{M}^{k}(a,b)\), \(I_{M}^{k}(a,b)\), and \(F_{M}^{k}(a,b)\) are described as follows._  
\[ \begin{array}{l}T_{M}^{k}(a,b)=\sup\left(T_{M}\left(a,a_{1}\right)\wedge T_{M }\left(a_{1},a_{2}\right)\wedge...\wedge T_{B}\left(a_{k-1},a_{k}\right)\right) \\ I_{M}^{k}(a,b)=\inf\left(I_{M}\left(a,a_{1}\right)\lor I_{M}\left(a_{1},a_{2} \right)\vee...\lor I_{M}\left(a_{k-1},a_{k}\right)\right)\\ F_{M}^{k}(a,b)=\inf\left(F_{M}\left(a,a_{1}\right)\lor F_{M}\left(a_{1},a_{2} \right)\vee...\lor F_{M}\left(a_{k-1},a_{k}\right)\right)\end{array} \]  
**Definition 6**. : _Let \(G=(N,M)\) be an SVNG. The strength of connection of a path \(P\) between two nodes \(a\) and \(b\) is defined by \(\left(T_{M}^{\infty}(a,b),I_{M}^{\infty}(a,b),F_{M}^{\infty}(a,b)\right)\) where:_  
\[ \begin{array}{l}T_{M}^{\infty}(a,b)=\sup\{T_{M}^{k}\left(a,b\right)\left|k=1, 2,...\right.\\ I_{M}^{\infty}(a,b)=\inf\{I_{M}^{k}\left(a,b\right)\left|k=1,2,...\right.\\ F_{M}^{\infty}(a,b)=\inf\{F_{B}^{k}\left(a,b\right)\left|k=1,2,...\right\}\end{array} \]  
**Definition 7**. : _Let \(G=(N,M)\) be an SVNG. \(G\) is said to be a connected SVNG [42] if there exists no isolated node in \(G\)._  
**Definition 8**. : _Let \(G=(N,M)\) be an SVNG and \(a\in V\) be node of \(G\). The degree of node \(a\) is the sum of the truth membership values, the sum of the indeterminacy membership values, and the sum of the membership values of falsity of all the arcs that are adjacent to node \(a\). The degree of node [35]\(a\) is denoted by \(d(a)=\left(d_{T}\left(a\right),d_{I}\left(a\right),d_{F}\left(a\right)\right)\) where:_  
\[d_{T}\left(a\right)=\sum_{ \begin{subarray}{c}a\in V\\ a\
eq b\end{subarray} }T_{M}\left(a,b\right),d_{I}\left(a\right)=\sum_{ \begin{subarray}{c}a\in V\\ a\
eq b\end{subarray} }I_{M}\left(a,b\right),d_{F}\left(a\right)=\sum_{ \begin{subarray}{c}a\in V\\ a\
eq b\end{subarray}