Neutrosophic Incidence Graphs With Application

In this research study, we introduce the notion of single-valued neutrosophic incidence graphs. We describe certain concepts, including bridges, cut vertex and blocks in single-valued neutrosophic incidence graphs. We present some properties of single-valued neutrosophic incidence graphs. We discuss the edge-connectivity, vertex-connectivity and pair-connectivity in neutrosophic incidence graphs. We also deal with a mathematical model of the situation of illegal migration from Pakistan to Europe.


Introduction
The concept of graph theory was introduced by Euler.A crisp graph shows the relations between the elements of the vertex set.A weighted graph gives the extent of these relations.Many problems can be solved if proper weights are given.However, in many situations, the weights may not known, and the relationship is uncertain.Hence, a fuzzy relation can be used to handle such situations.Rosenfeld [1] developed the concept of a fuzzy graph.He also discussed several concepts like edges, paths, bridges and connectedness in a fuzzy graph.Most of the theoretical development of fuzzy graph theory is based on Rosenfeld's initial work.Bhutani et al. [2,3] introduced the advance concepts in fuzzy graphs.
Sometimes when the relationship between the elements of the vertex set is indeterminate, the fuzzy graph and its extension fails.This indeterminacy can be overcome by using single-valued neutrosophic graphs [4].Dinesh, in [5], introduced the concept of unordered pairs of vertices which are not incident with end vertices.The fuzzy incidence graph not just shows the relations between vertices, but also provides information about the influence of a vertex on an edge.Dinesh extended the idea of the fuzzy incidence graph in [6] by introducing new concepts in this regard.Later, Methew et al. [7] discussed the connectivity concepts in fuzzy incidence graphs.Malik et al. [8] applied the notion of the fuzzy incidence graph in problems involving human trafficking.They discussed the role played by the vulnerability of countries and their government's response to human trafficking.Methew et al. [9] studied fuzzy incidence blocks and their applications in illegal migration problems.They used fuzzy incidence graphs as a model for a nondeterministic network with supporting links.They used fuzzy incidence blocks to avoid the vulnerable links in the network.
The paper is organized as follows: In Section 1, we give some preliminary notions and terminologies of fuzzy incidence graphs which are needed to understand the extended concept of the single-valued neutrosophic incidence graph.In Section 2, we present the definition of a single-valued neutrosophic incidence graph.We also discuss the edge-connectivity, vertex-connectivity and pair-connectivity in neutrosophic incidence graphs.In Section 3, we give a mathematical model of the situation of illegal migration from Pakistan to Europe.Finally, the paper is concluded by some remarks in Section 4. Below we present some preliminary definitions from [6] and [4].For further study on these topics, the readers are referred to references [1,[7][8][9][10][11][12][13][14][15][16].
Let G = (V, E) be a graph on a nonempty set, V.Then, G ′ = (V, E, I) is called an incidence graph, where I ⊆ V × E. The elements of I are called incidence pairs or simply, pairs.
A fuzzy incidence graph of an incidence graph, G ′ = (V, E, I), is an ordered-triplet, G = (µ, λ, ψ), where µ is a fuzzy subset of V, λ is a fuzzy relation of V, and ψ is a fuzzy subset of I such that We may compare elements of two neutrosophic sets A and B, that is

Single-Valued Neutrosophic Incidence Graphs Definition 1. A single-valued neutrosophic incidence graph of an incidence graph, G
A is a single-valued neutrosophic set on V.

2.
B is a single-valued neutrosophic relation on V.

3.
C is a single-valued neutrosophic subset of V × E such that Here, we discuss an example of a single-valued neutrosophic incidence graph (SVNIG).
Now we introduce the concepts of edge, pair, walk, trail, path and connectedness in an SVNIG.
In the above sequence, if all edges are distinct, then it is a trail, and if the pairs are distinct, then it is an incidence trail.P is called a path if the vertices are distinct.A path is called a cycle if the initial and end vertices of the path are same.Any two vertices of G are said to be connected if they are joined by a path.
Example 2. In the example presented earlier P 1 : a, (a, ab), ab, (b, ab), b, (b, bd), bd, (d, bd), d, (d, da), da, (a, da), a is a walk.It is a closed walk since the initial and final vertices are same, i.e., it is not a path, but it is a trail and an incidence trail.P 2 : a, (a, ab), ab, (b, ab), b, (b, bd), bd, (d, bd), d P 2 is a walk, path, trail and an incidence trail.
Definition 6.In an SVNIG, the strength of a path, P, is an ordered triplet denoted by S(P) = (s 1 , s 2 , s 3 ), where Similarly, the incidence strength of a path, P, in an SVNIG is denoted by Example 3. Let G = (V, E, I) be an incidence graph, as shown in Figure 3, and G = (A, B, C) is an SVNIG associated with G, which is shown in Figure 4.

S ∞ (l,m) is sometimes called the connectedness between l and m.
Similarly, the greatest incidence strength of the path from l to m, where l,m ∈ A * ∪ B * is the maximum incidence strength of all paths from l to m. IS ∞ (l, m) = max{IS(P 1 ), IS(P 2 ), 11 , is 12 , is 13 , ...), max(is 21 , is 22 , is 23 , ...), min(is 31 , is 32 , is 33 , ...) , where P j , j = 1, 2, 3, . . .are different paths from l to m. IS ∞ (l, m) is sometimes referred as the incidence connectedness between l and m.
Example 4. In the SVNIG given in Figure 4, the total paths from vertex u to w are as follows: So, G is a neutrosophic cycle.Furthermore, G is a neutrosophic incidence cycle since there is more than one pair, namely, (b, ab) and (d, de), such that The concepts of bridges, cutvertices and cutpairs in SVNIG are defined as follows.

Definition 11. Let G = (A, B, C) be an SVNIG. An edge, uv, in G is called a bridge if, and only if, uv is a bridge in G
where S ′ ∞ (x, y) and S ∞ (x, y) denote the connectedness between x and y in G ′ = G−{uv} and G, respectively.
An edge, uv, is called a neutrosophic incidence bridge if where IS ′ ∞ (x, y) and IS ∞ (x, y) denote the incidence connectedness between x and y in G ′ = G−{uv} and G, respectively.

in an SVNIG is called a neutrosophic cutvertex if the connectedness between any two vertices in G
′ = G−{v} is less than the connectedness between the same vertices in G-that is, S ′ ∞ (x, y) < S ∞ (x, y), for some x, y ∈ A * .
A vertex, v, in SVNIG G is a neutrosophic incidence cutvertex if for any pair of vertices, x, y, other than v, the following condition holds: where IS ′ ∞ (x, y) and IS ∞ (x, y) denote the incidence connectedness between x and y in G ′ = G−{v} and G, respectively.
Definition 13.Let G = (A, B, C) be an SVNIG.A pair (u, uv) is called a cutpair if, and only if, (u, uv) is a cutpair in G * = (A * , B * , C * )-that is, after removing the pair (u, uv), there is no path between u and uv.Let G = (A, B, C) be an SVNIG.A pair (u, uv) is called a neutrosophic cutpair if deleting the pair (u, uv) reduces the connectedness between u, uv ∈ A * ∪ B * , that is, where S ′ ∞ (u, uv) and S ∞ (u, uv) denote the connectedness between u and uv in G where IS ′ ∞ (u, uv) and IS ∞ (u, uv) denote the incidence connectedness between u and uv in Example 6.In the SVNIG, G, given in Figure 6, ab and bc are bridges, since their removal disconnects the underlying graph, G * .In G, ab, bc, cd and de are neutrosophic bridges, since, for a, e ∈ A * , after the removal of each of the bridges.The edges-ab, bc, cd and de-are neutrosophic incidence bridges in G as well.b and c are cutvertices.In addition, all the vertices of G are neutrosophic cutvertices, except for a, since the removal of a does not affect the connectedness of G. b, c, d and e are neutrosophic incidence cutvertices in G.
(c, 0.6, 0.6, 0.2) (d, 0.4, 0.5, 0. (e, 0.3, 0.2, 0.5) Proof.Let uv be a neutrosophic bridge and suppose, on the contrary, that uv is the weakest edge of a cycle.Then, in this cycle, we can find an alternative path, P 1 , from u to v that does not contain the edge uv, and S(P 1 ) is greater than or equal to S(P 2 ), where P 2 is the path involving the edge uv.Thus, removal of the edge uv from G does not affect the connectedness between u and v-a contradiction to our assumption.Hence, uv is not the weakest edge in any cycle.
Theorem 2. If (u, uv) is a neutrosophic incidence cutpair, then (u, uv) is not the weakest pair in any cycle.
Proof.Let (u, uv) be a neutrosophic incidence cutpair in G. On contrary suppose that (u, uv) is a weakest pair of a cycle.Then we can find an alternative path from u to uv having incidence strength greater than or equal to that of the path involving the pair (u, uv).Thus, removal of the pair (u, uv) does not affect the incidence connectedness between u and uv but this is a contradiction to our assumption that (u, uv) is a neutrosophic incidence cutpair.Hence (u, uv) is not a weakest pair in any cycle.
Proof.Let G be an SVNIG, and uv is a neutrosophic bridge in G. On the contrary, suppose that Then, there exists a u-v path, P, with for all edges on path P. Now, P, together with the edge, uv, forms a cycle in which uv is the weakest edge, but it is a contradiction to the fact that uv is a neutrosophic bridge.Hence, Proof.The proof is on the same line as Theorem 3.
Theorem 5. Let G = (A, B, C) be an SVNIG and G * = (A * , B * , C * ) is a cycle.Then, an edge, uv, is a neutrosophic bridge of G if, and only if, it is an edge common to two neutrosophic incidence cutpairs.
Proof.Suppose that uv is a neutrosophic bridge of G.Then, there exist vertices u and v with the uv edge lying on every path with the greatest incidence strength between u and v. Consequently, there exists only one path, P, (say) between u and v which contains a uv edge and has the greatest incidence strength.Any pair on P will be a neutrosophic incidence cutpair, since the removal of any one of them will disconnect P and reduce the incidence strength.Conversely, let uv be an edge common to two neutrosophic incidence cutpairs (u, uv) and (v, uv).Thus both (u, uv) and (v, uv) are not the weakest cutpairs of G. Now, G * = (A * , B * , C * ) being a cycle, there exist only two paths between any two vertices.Also the path P 1 from the vertex u to v not containing the pairs (u, uv) and (v, uv) has less incidence strength than the path containing them.Thus, the path with the greatest incidence strength from u to v is Therefore, uv is a neutrosophic bridge.It is not necessary for all edges and pairs to be strong.Edges and pairs exist which are not strong in an SVNIG.Such edges and pairs are given in the following definition.Definition 16.Let G = (A, B, C) be an SVNIG.An edge, uv, is said to be a δ-edge if (T B (uv), I B (uv), F B (uv)) < S ′ ∞ (u, v).
On the contrary, suppose that (u, uv) is not a strong incidence pair.Then, it follows that IS ′ ∞ (u, uv) > (T C (u, uv), I C (u, uv), F C (u, uv)).
Let P be the path from u to uv in G ′ = G−{(u, uv)} with the greatest incidence strength.Then, P together with (u, uv), forms a cycle in G. Now, in this cycle, (u, uv) is the weakest pair, but, based on

Definition 3 .
If xy ∈ B * , then xy is an edge of the SVNIG G = (A, B, C) and if (x, xy), (y, xy) ∈ C * , then (x, xy) and (y, xy) are called pairs of G. Definition 4. A sequence