Edge integrity for signed fuzzy graphs

Abstract

Transport network is the backbone of economy. Every path has some positive and negative attributes such as transportation cost, road condition, traveling time etc., These attribute values are taken as fuzzy membership value with either positive or negative sign when modeling the transport network as signed fuzzy graph. The stability of these type of signed fuzzy graphs are discussed with the help of vulnerability parameters and edge integrity. In this paper, we have introduced complete signed fuzzy graph, signed fuzzy star graph, complement of a signed fuzzy graph, union of two signed fuzzy graph, join of two signed fuzzy graph and cartesian product of two signed fuzzy graphs. For some standard signed fuzzy graph edge integrity value is calculated. Further this concept is applied in supply chain network with three layers, to study its stability and optimum path.

Keywords

Vulnerability parameters edge integrity signed fuzzy graph

1 Introduction

The economic growth of a country depends on the flow of resources from one place to another. It creates a transportation network around the world. Any network is a graph. These transportation network model can also be a type of graph, which deals with optimum route procedure in inventory. These type of graphs help to smooth flow of materials, availability and shortage, stability of network, connectivity issues and efficient routing. If a material is moved from one place x to another y, say via edge xy, it has some pros and cons. This type of issue can be analyzed by using signed graph. Signed graph is applied in the decision making problems. While passing through the edge, the profits are taken as positive sign (+) and loss are taken as negative sign (-) based on the road condition, transportation cost, toll rates and former’s profit. In a signed graph, all the vertices and edges are assigned, either + or - sign. Vulnerability parameter deals with the stability of the network. Among the vulnerability parameters, edge integrity deals with the impact removed edges on the remaining connected sub-graphs and general graph. The fuzzy graph with either positive or negative sign with its fuzzy membership value is a signed fuzzygraph.

Signed graph is introduced by Frank Harary in 1955 [10] with its basic terms. The standard types of signed graph are defined in [11]. T. Zaslavsky has introduced theory of orientation of graphs, matroids of signed graph and extended to signed colours in [44 –46]. Fred S.Roberts discussed the Balanced Signed Graphs in [30]. K. A. Germinaa et al. discussed the application of signed graphs in [8]. Homomorphism of signed graph has been introduced by Reza Naserasr et al. [16]. Nutan G. Nayak has introduced net regular signed graphs in [17]. F. Ramezani defined strongly regular for signed graph in [26]. Fuzzy graph concepts are introduced by [31] after defining fuzzy Sets and relations in [42, 43] some results and applications are given in [5–7 , 24]. Intutionestic Fuzzy Graphs are introduced by [19] properties and applications are given in [4, 36]. This work has been expanded to Bipolar Fuzzy graphs [1] results are discussed and applied in [20–23 , 29]. The concept of a vague graph was introduced in [25], some properties and applications are given by [27, 32]. This concept was developed rapidly to plithogenic set [37] as a generalisation of Neutrosophic set [40].

Chung-cheng Han et al. incorporated Fuzzy set theory with Signed Directed Graphs in [9]. The Signed Fuzzy Graph is introduced by Nirmala et al. [18]. Some basic definitions are redefined by S. N. Mishra et al. while defining intuitionistic fuzzy signed graphs [15]. Fuzzy signed Directed Graph used for Mathematical modelling [41], networking [14], root cause analysis [12] and grouping elastic energy [39].

The integrity is introduced as a vulnerability parameter by Barefoot. et al. [3]. The results are consolidated in the survey paper [2]. The edge integrity of graphs introduced in [13]. The concept of edge integrity of fuzzy graphs is discussed in [33 –35]. In the signed fuzzy graphs,various types of integrity are introduced and discussed by R. Sundareswaran et al. in [38]. In a signed fuzzy graph, the edge integrity is defined as minimum of sum of cardinality of edge subset and maximum order of largest connected component, while removing the edge subset.

2 Motivation

Edge integrity is defined for crisp graph, signed graph and fuzzy graph. Crisp graph deals with the stability but not optimality in signed graph. The advantage of fuzzy graph is that the linguistic variables can converted into membership values. But fuzzy edge integrity handles either profit or loss alone. All these drawbacks are overcome by edge integrity of signed fuzzy graphs this parameter help us to find a stable and optimal route dealing with profit and loss.

In this, section 1 gives a brief introduction and literature survey of the work. The importance of the parameter is discussed in section 2. Section 3 deals with basic definitions used in this article. A comparison of graphs is also tabulated in the section. Signed fuzzy graphs and all other basic definitions are defined in Section 4. Edge integrity results are discussed in section 5. Edge integrity for union, join and cartesian product of signed fuzzy graph are also included. The concept of edge integrity is applied in a supply chain network for Paddy for signed fuzzy graph and it is presented in section 6. The last section concludes with results and findings.

3 Preliminaries

The crisp graph G=(V,E) is defined as a nonempty set of vertices V and a edge set E which is a subset of V×V. The basic terminologies in crisp graph are taken from Rosenfield [31]. The fuzzy graph G : (σ, μ) is defined such that σ : V → [0, 1] and μ (x, y) : E → [0, 1] with μ (x, y) ≤ σ (x) ∧ σ (y) , ∀ (x, y) ∈ E. If μ (x, y) = σ (x) ∧ σ (y) then the edge is known as strong edge of the fuzzy graph G. In the strong fuzzy graph G every edge is strong edge. A fuzzy graph G : (σ, μ) is called a fuzzy complete graph if every edge is strong edge and the underlying crisp graph is a complete fuzzy graph. The vertex cardinality is the sum of all vertex membership value of adjacent vertices. The order of a fuzzy graph is a sum of all vertex membership value of the graph. Similarly, size of a fuzzy graph is the sum of all edge membership value of the graph.

The signed fuzzy graph is a fuzzy graph with its vertex and edge membership value includes either + or - sign. Formally, it is defined as the signed fuzzy graph Σ_± = (σ, μ) such that σ : V → [-1, + 1] and μ : E → [-1, + 1] with μ (x, y) ≤ minimum {σ (x) , σ (y)} , ∀ (x, y) ∈ E. Here Σ is underlying fuzzy graph with the crisp graph G having vertex set V and edge set E.

Comparison of crisp, signed crisp graph, fuzzy graph and signed fuzzy graph is given in Table 1.

Table 1
Comparison

Crisp Graph Signed Crisp Graph Fuzzy Graph Signed Fuzzy Graph

Vertex and edge has membership value one With the underlying crisp graph, vertex and edge have either “+” or “-” sign The vertex membership value lies in [0, 1] and edge takes the membership value of same or less than of the adjacent vertices A fuzzy graph with “+” or “-” sign in vertex and edge

Always the membership is fixed Nature of the problem defines the sign of vertex and edge With same underlying crisp graph, by changing its membership value, we may generate infinitely many fuzzy graph Sign in the fuzzy graph is defined by linguistic terms

Removing an edge or a vertex forms a sub-graph The pros and cons depends on the sign of vertex and edge We may consider the graph itself fuzzy. For example, depends on time some streets are included and excluded in road transport network. At that time some edge are excluded and included where the fuzzy graph remains same The pros and cons depends on the sign of vertex and edge in fuzzy graphs

All the vertex and edges have equal importance The sign defines the importance of the vertex and edge Vertex membership value may be crisp and edge may be fuzzy or edge membership value may be crisp and vertex may be fuzzy or both are fuzzy Importance related to the underlying fuzzy graph

Any network is a model of a graph The sign enrich the importance of the vertex and sign Any network assigned with linguistic terms is a model of a fuzzy graph The results is more relevant to the linguistic term defined in the fuzzy graphs

Crisp Graph	Signed Crisp Graph	Fuzzy Graph	Signed Fuzzy Graph
Vertex and edge has membership value one	With the underlying crisp graph, vertex and edge have either “+” or “-” sign	The vertex membership value lies in [0, 1] and edge takes the membership value of same or less than of the adjacent vertices	A fuzzy graph with “+” or “-” sign in vertex and edge
Always the membership is fixed	Nature of the problem defines the sign of vertex and edge	With same underlying crisp graph, by changing its membership value, we may generate infinitely many fuzzy graph	Sign in the fuzzy graph is defined by linguistic terms
Removing an edge or a vertex forms a sub-graph	The pros and cons depends on the sign of vertex and edge	We may consider the graph itself fuzzy. For example, depends on time some streets are included and excluded in road transport network. At that time some edge are excluded and included where the fuzzy graph remains same	The pros and cons depends on the sign of vertex and edge in fuzzy graphs
All the vertex and edges have equal importance	The sign defines the importance of the vertex and edge	Vertex membership value may be crisp and edge may be fuzzy or edge membership value may be crisp and vertex may be fuzzy or both are fuzzy	Importance related to the underlying fuzzy graph
Any network is a model of a graph	The sign enrich the importance of the vertex and sign	Any network assigned with linguistic terms is a model of a fuzzy graph	The results is more relevant to the linguistic term defined in the fuzzy graphs

Definition 1. The vertex integrity [34] of fuzzy graph Σ : (σ, μ) is defined as $\tilde{I} (Σ) = \min {| S | + m (Σ - S) / S \subseteq V}$ where |S| is vertex cardinality of vertex subset S and m (Σ - S) is the order of the largest component of Σ - S.

Definition 2. The edge integrity [33] of fuzzy graph Σ : (σ, μ) is defined as $\tilde{I^{'}} (Σ) = \min {| S | + m (Σ - S) / S \subset E}$ , where |S| denotes sum of edge membership values of the edge subset S and m (Σ - S) denotes the maximum order of connected components after removing the set S.

Definition 3. The vertex integrity [38] of signed fuzzy graph Σ_± : (σ, μ) is defined as $\tilde{I_{\pm}} (Σ_{\pm}) = \min {| S | + m (Σ_{\pm} - S) / S \subseteq V}$ , where |S| is vertex cardinality and m (Σ_± - S) is the order of the largest component of Σ_± - S.

Definition 4. Let Σ_± : (σ, μ) be a signed fuzzy graph. The edge integrity of Σ is defined as $\tilde{I_{\pm}^{'}} (Σ_{\pm}) = min_{S \subset E} {| S | + m (Σ - S)}$ where |S| = ∑_e∈Sμ (E) and m (Σ - S) = ∑_v∈(Σ-S)μ (e).

Example 1. Let Σ : (σ, μ) be a signed fuzzy graph with vertex set V = {a, b, c} and edge set E = {(a, b) , (b, c) , (a, c)}. The membership values and its associate sign values are shown in Fig. 1 for that graph the edge integrity value is -0.3 the calculation part is shown in Table 2.

Fig. 1

Signed fuzzy graph.

Table 2

Calculation of edge integrity of signed fuzzy graph and the edge integrity is -0.3.

S	\|S\|	m (G - S)	\|S\| + m (G - S)
{ab}	-0.3	0	-0.3
{bc}	-0.3	0	-0.3
{ca}	-0.1	0	-0.1
{ab, bc}	-0.6	0.3	-0.3
{bc, ca}	-0.4	0.4	0
{ca, ab}	-0.4	0.1	-0.3
{ab, bc, ca}	-0.7	0.4	-0.3

4 Some definitions on signed fuzzy graphs

The following definitions are introduced with some examples.

Definition 5. In a complete fuzzy graph if every vertex and edge has a sign then it is called complete signed fuzzy graph.

Definition 6. A signed fuzzy star graph K_1,n has two vertex sets V and U with |V|=1 and |U| = n such that μ (v, u_i) ≠ 0 and μ (u_i, u_i+1) = 0, 1 ≤ i ≤ n.

Example 2. Let Σ : (σ, μ) be a signed fuzzy star graph with vertex set V = {a, b, c, d} and edge set E = {(a, b) , (a, c) , (a, d)}. The membership values and its associate sign values are shown in Fig. 2 for that graph the edge integrity value is 0.5 the calculation part is shown in Table 3.

Fig. 2

Example for signed fuzzy star graph.

Table 3

Calculation of edge integrity of signed fuzzy star graph and the edge integrity is 0.5.

S	\|S\|	m (G - S)	\|S\| + m (G - S)
{ab}	-0.2	0.7	0.5
{ac}	-0.1	0.6	0.5
{ad}	0.3	0.5	0.8
{ab, ac}	-0.3	0.8	0.5
{ac, ad}	0.2	0.5	0.7
{ab, ac, ad}	0	0.5	0.5

Definition 7. The Complement of a signed fuzzy graph G = (σ, μ) is a signed fuzzy graph $\bar{G} = (\bar{σ}, \bar{μ})$ where $\bar{σ} = σ$ and $\bar{μ} (u, v) = σ (u) \land σ (v) - μ (u, v)$ for all u, v in V.

Definition 8. A Signed Fuzzy graph G = (σ, μ) is Self-Complementary if $G = \bar{\bar{G}}$ .

Definition 9. Let G₁ = (σ₁, μ₁) and G₂ = (σ₂, μ₂) be the two signed fuzzy graphs. Then the union of two signed fuzzy graphs G₁ and G₂ is a signed fuzzy graph G = G₁ ∪ G₂ : (σ₁ ∪ σ₂, μ₁ ∪ μ₂)defined by

$σ_{1} \cup σ_{2} = {\begin{matrix} σ_{1} (u) & if u \in V_{1} - V_{2} \\ σ_{2} (u) & if u \in V_{2} - V_{1} \\ σ_{1} (u) \lor σ_{2} (u) & if u \in V_{1} \cap V_{2} \end{matrix}$ and $μ_{1} \cup μ_{2} = {\begin{matrix} μ_{1} (u, v) & if (u, v) \in E_{1} - E_{2} \\ μ_{2} (u, v) & if (u, v) \in E_{2} - E_{1} \\ μ_{1} (u, v) \lor μ_{2} (u, v) & if (u, v) \in E_{1} \cap E_{2} ∥ \end{matrix}$

Example 3. Let G₁ = (σ₁, μ₁) and G₂ = (σ₂, μ₂) be two complete signed fuzzy graphs with σ₁ (a) = 0.3, σ₁ (b) = 0.5, σ₁ (c) = -0.3 and σ₂ (d) = 0.2, σ₂ (e) = - o . 1, σ₂ (f) = 0.3, σ₂ (g) = 0.6. Then the union of two signed fuzzy graphs is shown in Fig. 4.

Fig. 3

Self-Complementary of a signed fuzzy graph.

Fig. 4

Union of two signed fuzzy graphs.

Definition 10. Let join of G₁ and G₂ is defined as a signed fuzzy graph if G = G₁ + G₂ = (σ₁ + σ₂, μ₁ + μ₂) on G^* = (V, E) with V₁ ∩ V₂ ≠ φ where V = V₁ ∪ V₂ and E = E₁ ∪ E₂ ∪ E′ Where E′ is the set of all edges joining vertices of V₁ with vertices of V₂ having (σ₁ + σ₂) (u) = (σ₁ ∪ σ₂) (u) for all u ∈ V₁ ∪ V₂ and $μ_{1} + μ_{2} = {\begin{matrix} (μ_{1} \cup μ_{2}) (u, v) & \in if (u, v) \in E_{1} \cup E_{2} \\ σ_{1} (u) \land σ_{2} (v) & \in if (u, v) \in E^{'} \end{matrix}$ .

Example 4. Let G₁ = (σ₁, μ₁) be a signed fuzzy graph with σ₁ (a) = 0.3, σ₁ (b) = -0.2 and μ₁ (a, b) = -0.2. Let G₂ = (σ₂, μ₂) be a signed fuzzy graph with σ₂ (c) = 0.5, σ₂ (d) = -0.4, σ₂ (e) = 0.7 and μ₂ (c, d) = -0.4, μ₂ (d, e) = -0.4. Then the join of two signed fuzzy graphs is shown in Fig. 5.

Fig. 5

Join of two signed fuzzy graphs.

Definition 11. The cartesian product of two signed fuzzy graphs G₁ and G₂ is defined as a signed fuzzy graph G = G₁ × G₂ = (σ₁ × σ₂, μ₁ × μ₂) on G^* = (V, E) where V = V₁ × V₂ and E = (u₁, u₂) (v₁, v₂) ifu₁ = v₁, (u₂, v₂) ∈ E₂oru₂ = v₂, (u₁, v₁) ∈ E₁.

Example 5. Let G₁ = (σ₁, μ₁) be a signed fuzzy graph with σ₁ (a) = 0.6, σ₁ (b) = -0.2 and μ₁ (a, b) = -0.2. Let G₂ = (σ₂, μ₂) be a signed fuzzy graph with σ₂ (c) = 0.4, σ₂ (d) = -0.3, σ₂ (e) = 0.5 and μ₂ (c, d) = -0.3, μ₂ (d, e) = -0.3. Then the cartesian product of two signed fuzzy graphs are shown in Fig. 6.

Fig. 6

Cartesian product of two signed fuzzy graphs.

5 Results on edge integrity of signed fuzzy graphs

The novel concept of the edge integrity for signed fuzzy graph is explored for some of the standard signed fuzzy graphs, union, join and cartesian product of signed fuzzy graphs.

Theorem 1. Let Σ_± be a complete signed fuzzy graph having all vertex membership values (and hence edges) -ve. Then the edge integrity is $I_{\pm}^{'} (Σ_{\pm})$ =size of Σ_± + σ_n, where σ_n = max (σ).

Proof. By the definition, the edge integrity is defined as $I_{\pm}^{'} (Σ_{\pm}) = \min {| S | + m (Σ_{\pm} - S)}$ where S⊆E. Let σ₁, σ₂, …, σ_n be the vertex membership values such that σ₁ < σ₂ < … < σ_n. Given that Σ_± is complete and also sign of vertex and edge membership is negative. Then the graph has (n - 1) σ₁edges,(n - 2) σ₂ edges,…,1 . σ_n-1 edges. Since all the fuzzy values are negative and to obtain minimality, all the edges should be included in edge integrity set S. Thus, the edge integrity contains all the edges. This implies |S|=size of the graph. Then Σ_± - S becomes a set of isolated vertices and m (Σ_± - S) takes the value σ_n. Hence $I_{\pm}^{'} (Σ_{\pm})$ =size of Σ_±+σ_n, where σ_n = max (σ). □

Example 6. By theorem 1 the edge integrity for figure 7 is -0.9.

Fig. 7

Example for theorem 1.

Theorem 2. Let Σ_± be a complete signed fuzzy graph having all vertex membership values (edges also) positive. Then the edge integrity is $I_{\pm}^{'} (Σ_{\pm})$ =sum of all (σ_n) , ∀ n.

Proof. If all the edge and vertex membership values are positive, then the complete signed fuzzy graph becomes, complete fuzzy graph. By [Theorem 4.3] in [33], the edge integrity of complete fuzzy graph is order of fuzzy graph, hence the theorem. □

Example 7. Using theorem 2 in Fig. 8 the edge integrity of signed fuzzy graph is 1.

Fig. 8

Example for theorem 2.

Theorem 3. Let Σ_± be a complete signed fuzzy graph with n vertices in which k membership values are negative then $I_{\pm}^{'} (Σ_{\pm}) = (n - 1) σ_{1} + (n - 2) σ_{2} + \dots, + (n - k) σ_{k} + σ_{k + 1} + σ_{k + 2} + \dots + σ_{n}$ . i.e., $I_{\pm}^{'} (Σ_{\pm}) = (n - 1) σ_{1} + (n - 2) σ_{2} + \dots + (n - k) σ_{k} + order (Γ_{\pm})$ . where Γ_± is a subgraph of Σ_± having only (n - k) positive vertex membership values.

Proof. Let Σ_± be a complete signed fuzzy graph with n vertices in which σ₁ ≤ σ₂ ≤ … ≤ σ_k are negative vertex membership values and σ_k+1 ≤ σ_k+2 ≤ … ≤ σ_n are positive vertex membership values. By including any -ve edge and by excluding +ve edge in edge integrity set we may get minimum. In the given graph there are (n - 1) edges with vertex membership value σ₁, (n - 2) edges with vertex membership value σ₂, …,(n - k) edges with vertex membership value σ_k. All the remaining edges are with positive membership values. For minimality all negative membership edges are included. For the given edge integrity set includes (n - 1) σ₁ + (n - 2) σ₂ + … + (n - k) σ_k edges which have negative edge membership values. At this stage the graph is decomposed into k negative isolated vertices and a Γ_± complete positive graph with (n - k) vertices which all have positive membership values. m (Σ_±) becomes order of Γ_±. $I_{\pm}^{'} (Σ_{\pm}) = (n - 1) σ_{1} + (n - 2) σ_{2} + \dots + (n - k) σ_{k} + order (Γ_{\pm})$ . □

Example 8. Using theorem 3 in Fig. 1 the edge integrity is -0.3.

Corollary 1. For the above theorem if the graph Σ_± is decomposed into two complete signed fuzzy graphs M_± and Γ_±, whose all vertex membership values are +ve and -ve respectively then I′ (Σ_±) = size (M_±) + order (Γ_±).

Theorem 4. Let Σ_± be a complete signed fuzzy graph with n vertices. The edge integrity of Σ_± is $I_{\pm}^{'} (Σ_{\pm}) = (n - 1) σ_{1} + I_{\pm}^{'} (Σ_{\pm} - v)$ , where σ₁ = min (σ) and v be the vertex such that σ (v) = σ₁.

Proof. By induction property, we prove the theorem.

For n = 2

Let v and w be the two vertices such that σ (v) = σ₁, σ (w) = σ₂ without loss of generality σ₁ ≤ σ₂. The edge membership value μ₁ = σ₁, by the definition of complete signed fuzzy graph. If S = φ, then |S| + m (Σ_± - v) = 0 + σ₁ + σ₂ = σ₁ + σ₂. If S = μ₁ (= E), then |S| + m (∑_± - v) = μ₁ + σ₂ = σ₁ + σ₂, $I_{\pm}^{'} (Σ_{\pm}) = σ_{1} + σ_{2}$ . Now $(n - 1) σ_{1} + I_{\pm}^{'} (Σ_{\pm} - v) = (2 - 1) σ_{1} + σ_{2} = σ_{1} + σ_{2}$ , $I_{\pm}^{'} (Σ_{\pm}) = (n - 1) σ_{1} + I_{\pm}^{'} (Σ_{\pm} - v)$ . Thus, the statement is true for n = 2.

Let us assume that the result is true for n = k. i.e., $I_{\pm}^{'} (Σ_{\pm}) = (k - 1) σ_{1} + I_{\pm}^{'} (Σ_{\pm} - v)$

For n = k + 1,

Now define a new complete signed fuzzy graph Z_± which includes one more vertex u, that is adjacent to all the vertex of Σ_±. Thus Z_± is a complete signed fuzzy graph with k + 1 vertices.

If σ (w) > σ₁, then by [Theorem.3] $I_{\pm}^{'} (Z_{\pm}) = k σ_{1} + (k - 1) σ_{2} + \dots + order (Σ_{\pm}) = k σ_{1} + I_{\pm}^{'} (Z_{\pm} - v)$ .

If σ (w) ≤ σ₁, then by above [Theorem 3] $I_{\pm}^{'} (Z_{\pm}) = k σ_{1} + (k - 1) σ_{2} + \dots + order (Σ_{\pm}) = k σ_{1} + I_{\pm}^{'} (Z_{\pm} - v)$ . Thus, the result is true for n = k + 1. Hence by induction principle, $I_{\pm}^{'} (Σ_{\pm}) = (n - 1) σ_{1} + I_{\pm}^{'} (Σ_{\pm} - v)$ . □

Theorem 5. Let Σ_± be a complete star signed fuzzy graph with n vertices. The edge integrity of Σ_± is $I_{\pm}^{'} (Σ_{\pm}) =$ size of the graph G+σ_n, where σ_n = max (σ).

Proof. In a star K_1,n, there are n edges. Including any edge in the edge integrity set S, we get a single connected graph. Since every edge has the membership value which is minimum between adjacent vertices, the edge integrity includes all n edges which disconnect all the vertices. At this stage m (G - S) = maximum. $I_{\pm}^{'} (Σ_{\pm}) =$ size of graph G + max of σ. □

Example 9. By using theorem 5, the edge integrity for Fig. 2 is 0.5.

Theorem 6. Let Σ_± and Z_± be two signed fuzzy graphs, whose underlying crisp graphs are G₁ = (V₁, E₁) and G₂ = (V₂, E₂) with V₁ ∩ V₂ = φ then $I_{\pm}^{'} (Σ_{\pm} \cup Z_{\pm}) = | S_{1} |_{e} + | Z_{\pm} |_{v} or | S_{2} |_{e} + | Σ_{\pm} |_{v} or I_{\pm}^{'} (Z_{\pm}) or I_{\pm}^{'} (Σ_{\pm})$ , where S₁ and S₂ are edge sets of G₁ and G₂.

Proof. In the given graphs there is no common vertex between Σ_± and Z_±. Therefore, their union consists of two disconnected components.

Case 1: If any of the graph has no edges say Σ_±, then the edge integrity value equals to another graph Z_±. In this case $I_{\pm}^{'} = \min (I_{\pm}^{'} (Σ_{\pm}), I_{\pm}^{'} (Z_{\pm}))$ .

Case 2: Let S₁ be the edge integrity set of Σ_± and S₂ be the edge integrity set of Z_± if |S₁| < |S₂| and m (Σ_± - S₁) < |Z_±| then $I_{\pm}^{'} (Σ_{\pm} \cup Z_{\pm}) = | S_{1} |_{e} + | Z_{\pm} |_{v}$ if |S₁| < |S₂| and |Z_±| < m (Σ_± - S₁) then $I_{\pm}^{'} (Σ_{\pm} \cup Z_{\pm}) = I_{\pm}^{'} (\sum_{\pm})$ .

considering the above cases $I_{\pm}^{'} (Σ_{\pm} \cup Z_{\pm}) = | S_{1} |_{e} + | Z_{\pm} |_{v} or | S_{2} |_{e} + | Σ_{\pm} |_{v} or I_{\pm}^{'} (Z_{\pm}) or I_{\pm}^{'} (\sum_{\pm})$ . □

Theorem 7. Let Σ_± and Z_± be two signed fuzzy graphs with underlying crisp graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂) with V₁ ∩ V₂ ≠ φ, V₁ ⊄ V₂,V₂ ⊄ V₁, $I_{\pm}^{'} (Σ_{\pm} \cup Z_{\pm}) = {| S_{1} |_{e} + | S_{2} |_{e}} + m [(Σ_{\pm} \cup Z_{\pm}) - (S_{1} \cup S_{2})]$ , where S₁ and S₂ are edge sets of G₁ and G₂.

Proof. To disconnect the graph Σ_± ∪ Z_±, remove the edge sets S₁ and S₂, we get the maximum connected component (Σ_± ∪ Z_±) - (S₁ ∪ S₂). Any other edge set contradicts with the edge integrity value of Σ_± and Z_±. □

Corollary 2. Let Σ_± be a signed fuzzy graph V₁ ∩ V₂ = V₁orV₂ then $I_{\pm}^{'} (Σ_{\pm} \cup Z_{\pm}) = I_{\pm}^{'} (Σ_{\pm}) or I_{\pm}^{'} (Z_{\pm})$ .

Proof. If V₁ ∩ V₂ = V₁ then Σ_± ∪ Z_± = Σ_±, $I_{\pm}^{'} (Σ_{\pm} \cup Z_{\pm}) = I_{\pm}^{'} (Σ_{\pm})$

similarly V₁ ∩ V₂ = V₂ then Σ_± ∪ Z_± = Z_±, $I_{\pm}^{'} (Σ_{\pm} \cup Z_{\pm}) = I_{\pm}^{'} (Z_{\pm})$ □

Theorem 8. Let Σ_± and Z_± be two signed fuzzy graphs then $I_{\pm}^{'} (Σ_{\pm} + Z_{\pm}) = | E^{'} | + I_{\pm}^{'} (Σ_{\pm} \cup Z_{\pm})$ . □

Proof. By definition of join, E′ be the set of edges linking every vertex of Σ_± with every vertex of Z_±. So to disconnect Σ_± + Z_±, we include all the edges E′ in edge integrity set. At this stage, we get two disconnected components Σ_± and Z_±. Now the edge integrity implies union of Σ_± ∪ Z_±. $I_{\pm}^{'} (Σ_{\pm} + Z_{\pm}) = | E^{'} | + I_{\pm}^{'} (Σ_{\pm} \cup Z_{\pm})$ . □

Theorem 9. Let Σ_± be signed fuzzy graphs with crisp graph K₂ and Z_± be any signed fuzzy graph then $I_{\pm}^{'} (Σ_{\pm} \times Z_{\pm}) = | E^{'} | + I_{\pm}^{'} (\sum_{\pm} \cup Z_{\pm})$ .

Proof. By the definition of cartesian product if K₂ and Z_±. E′ be the set of edges connecting the vertices of Z_± with a duplicate of Z_±. So to disconnect Σ_± × Z_± at least we have to remove E′. Then the remaining graph is a graph with two disconnected component with no common vertex. Thus the integrity becomes union of two disjoint graphs which implies $I_{\pm}^{'} (Σ_{\pm} \times Z_{\pm}) = | E^{'} | + I_{\pm}^{'} (Σ_{\pm} \cup Z_{\pm})$ . □

In a crisp graph, maximum order of a component purely depends on the number of vertices in that component whereas in a signed graph, the membership values contribute more in deciding the maximum order. For example, consider a component of (n > 1) vertices having all negative membership values and a single vertex with positive membership values. In a crisp sense the component with n vertices is maximum where as in fuzzy sense the single vertex is the maximum order. Based on this we have the following remarks.

•Let Σ_± be a signed fuzzy graph with n vertices having σ₁ = min (σ) and σ_n = max (σ).

Remark 1. The edge integrity of Σ_± whose underlying crisp graph is a path P_n has bounds $(⌈ 2 \sqrt{n} ⌉ - 1) σ_{1} \leq I^{'} (Σ_{\pm}) \leq (⌈ 2 \sqrt{n} ⌉ - 1) σ_{n}$ .

Remark 2. The edge integrity of Σ_± whose underlying crisp graph is a cycle C_n has bounds $(⌈ 2 \sqrt{n} ⌉) σ_{1} \leq I^{'} (Σ_{\pm}) \leq (⌈ 2 \sqrt{n} ⌉) σ_{n}$ .

6 Application

Agriculture is the backbone of the Indian economy. The majority of Indians who belong to the rural area have agriculture as their primary occupation. Agriculture mainly depends on the climate, water, land and transportation. The economically suppressed Indian farmers are maximum from lower-income to middle income. Hence, selling the cultivated products at the earliest is the only option left for two reasons. First, the demand for logistics causes a delay in shifting the food grains to the markets. Next, they do not have adequate storage facilities. These two factors force the poor farmers to sell the cultivated to the local vendors for a low cost to repay the loans obtained to purchase seeds, fertilisers, and other agriculture needs. Even though government offers agriculture loans, the hidden risk factors or the question of repaying capacity prevents the farmers from availing of bank loans. Using this gap between the government plans and the beneficiaries leads the farmers to avail loans from the local vendors and greedy merchants for high-interest rates. This is also one of the factors that drive the cultivators to sell the cultivated to the local vendors, and this causes the mediators to fix the cost for food grains with high-profit margins. There is a serious discussion on the agriculture supply chain and the reasons, business method, implementing digital and analytics technologies. Traders using the digital twin technique may gain an advantage in the highly competitive market. The agriculture supply chain gets complicated by disjointed inbound and outbound networks. A typical agricultural supply chain consists of three steps: farmers to intermediate silos, silos to transformation plants, and transformation plants to clients. The supply chain problem using fuzzy graphs could help to overcome the crisis in the transportation network. Experts define the membership value of the vertex based on the average transformation costs, Silo-management Costs, and cost of the food grains. If the value is more than the average price, the fuzzy value is positive. Similarly, if it is below the average price, the fuzzy value is negative. Edge membership values are defined based on the distance, road condition and road tolls. All the deciding parameters of profit and loss may include while defining the membership values of vertex and edge.

Consider the network with three layers from farms to processing plant and processing plant to retailers. These network can be modeled as signed fuzzy graph. The networks nodes refers farms (F1, F2, F3), processing plant (P1, P2, P3), and retailers (R1, R2, R3). Now, to find the stability of the network and optimum path from farmers to retailers, the edge integrity of signed fuzzy graphs is used.

Table 4
From the table, the edge integrity is 0.4.

S |S| m (G - S) |S| + m (G - S)

{F₁P₁, P₁M₁} 0.4 0.4 0.8

{F₂P₂, P₂M₂} 0 0.4 0.4

{F₃P₃, P₃M₃} 0.7 0.3 1

{F₁P₂, P₂M₃} 0.1 0.4 0.5

{F₂P₃, P₃M₃} 0.5 0.4 0.9

S	\|S\|	m (G - S)	\|S\| + m (G - S)
{F₁P₁, P₁M₁}	0.4	0.4	0.8
{F₂P₂, P₂M₂}	0	0.4	0.4
{F₃P₃, P₃M₃}	0.7	0.3	1
{F₁P₂, P₂M₃}	0.1	0.4	0.5
{F₂P₃, P₃M₃}	0.5	0.4	0.9

Fig. 9

Application.

From Fig. 9, we have 27 paths from farmer to retailers. The impact of all these paths on the network is studied through edge integrity calculation. Some paths which lead to the calculation are given in the table 3. Based on these calculation, the paths F₂P₂, P₂M₂ may give a good profit in paddy business as the other paths have more vulnerable.

7 Conclusion

In this paper, edge integrity of signed fuzzy graph is defined and explained with an example. This novel concept is explored for complete signed fuzzy graphs with all the possibilities of sign and complete star signed fuzzy graph. For union, join and cartesian product of signed fuzzy graphs, edge integrity value is calculated. The complement of signed fuzzy graph and self complementary signed fuzzy graph is defined and explained with example. The edge integrity of signed fuzzy graph is discussed with a practical application in supply chain management. The bounds for path and cycle are discussed. This work can be further extended to Intutionsistic Fuzzy Graphs, Bipolar Fuzzy Graphs and Neutrosophic Graphs.

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Edge integrity for signed fuzzy graphs

Abstract

Keywords

1 Introduction

2 Motivation

3 Preliminaries

Table 4 From the table, the edge integrity is 0.4. S |S| m (G - S) |S| + m (G - S) {F1P1, P1M1} 0.4 0.4 0.8 {F2P2, P2M2} 0 0.4 0.4 {F3P3, P3M3} 0.7 0.3 1 {F1P2, P2M3} 0.1 0.4 0.5 {F2P3, P3M3} 0.5 0.4 0.9

References

Table 4
From the table, the edge integrity is 0.4.

S |S| m (G - S) |S| + m (G - S)

{F₁P₁, P₁M₁} 0.4 0.4 0.8

{F₂P₂, P₂M₂} 0 0.4 0.4

{F₃P₃, P₃M₃} 0.7 0.3 1

{F₁P₂, P₂M₃} 0.1 0.4 0.5

{F₂P₃, P₃M₃} 0.5 0.4 0.9