## 2 Materials and Methods
### Basic Definitions  
**Definition 1** ([39]).: _A digraph consists of two types of elements: vertices and arcs. Each arc connects two vertices in a certain direction._  
For example, the digraph below (Figure 1) contains three vertices \(\{a,\,b,\,c\}\) and three arcs \(\{1,\,2,\,3\}\). Arcs 1 and 2 connect vertices \(a\) and \(b\), while arc 3 connects vertices \(c\) and \(a\).  
**Definition 2** ([39]).: _Let \(D\) be a digraph with \(n\) vertices (1, 2, 3, etc.). The adjacency or connection matrix \(A(D)\) of \(D\) is an \(n\times n\) matrix with entries in row \(i\) and column \(j\) representing the number of arcs from vertex \(i\) to vertex \(j\)._  
For example, given the digraph shown above in Figure 1 with three labelled vertices, we obtain the following matrix with three rows and three columns. The numbers appearing in the matrix refer to the number of arcs joining the corresponding vertices in the digraph (Figure 2).  
Figure 1: A simple digraph.  
**Definition 3** ([29]).: _In neutrosophic logic, each logical variable \(x\) is described by an ordered triple \(x=(T,\ I,\ F)\), where \(T\) is the degree of truth, \(F\) is the degree of falsehood, and \(I\) is the level of indeterminacy._  
**Definition 4** ([29]).: _A neutrosophic graph is one in which at least one edge is an indeterminacy, indicated by dotted lines._ For example, the following are neutrosophic graphs (Figure 3):  
**Definition 5** ([29]).: _A neutrosophic directed graph extends a neutrosophic graph by ensuring that at least one edge is not only indeterminate but also directed, meaning it captures both uncertainty and a directional relationship between nodes._  
**Definition 6** ([29]).: _An NCM is a neutrosophic directed graph with nodes representing policies, occurrences, and so on, and edges representing causalities or indeterminates. It expresses the causal link between ideas._  
Let \(C_{i}\) and \(C_{j}\) represent the two nodes of the NCM. The directed edge from \(C_{i}\) to \(C_{j}\) represents the causation of \(C_{i}\) on \(C_{j}\), also known as the link. Each edge in the NCM is weighted with a number from the set \(\{-1,0,1,I\}\). Let \(e_{ij}\) represent the weight of the directed edge \(C_{i}C_{j}\), where \(e_{ij}\in\{-1,\ 0,\ 1,\ I\}\). If \(C_{i}\) has no influence on \(C_{j}\), then \(e_{ij}=0\), (ii) \(e_{ij}=1\) if an increase (or decrease) in \(C_{i}\) produces an increase (or decrease) in \(C_{j}\); (iii) \(e_{ij}=-1\) if an increase (or decrease) in \(C_{i}\) causes a decrease (or increase) in \(C_{j}\); and (iv) \(e_{ij}=I\) if the relationship or impact of \(C_{i}\) on \(C_{j}\) is unclear [29].  
**Definition 7** ([29]).: _Let \(G\) denote a neutrosophic graph. The adjacency matrix of \(G\) with entries from the set (I, 0, 1) is known as the graph's neutrosophic adjacency matrix (Figure 4)._  
Figure 4: Neutrosophic graph and its adjacency matrix [27].  
Figure 3: Simple neutrosophic graphs.  
**Definition 8 ([29]).** (Synthesizing of different NCMs).: _A finite number of NCMs can be combined together to produce the joint effect of all NCMs. If \(N(E_{1})\), \(N(E_{2})\),..., \(N(E_{p})\) be the neutrosophic adjacency matrices of an NCM with nodes \(C_{1}\), \(C_{2}\),, \(C_{n}\) then the combined NCM is obtained by adding all the neutrosophic adjacency matrices \(N(E_{1})\),, \(N(E_{p})\). We denote the combined NCMs adjacency neutrosophic matrix by the following_  
\[N(E)=N(E_{1})+N(E_{2})+\ldots+N(E_{p}) \tag{1}\]