Fuzzy soft graphs with applications

Abstract

A fuzzy soft set is a mapping from parameter set to the fuzzy subsets of universe. Fuzzy soft set theory provides a parameterized point of view for uncertainty modeling and soft computing model. In this research article, we apply the concept of fuzzy soft sets to graphs. We present the concept of fuzzy soft graphs, various methods of their construction, and investigate some of their related properties. We discuss certain types of irregular fuzzy soft graphs. We also describe applications of fuzzy soft graphs in social network and road network.

Keywords

Fuzzy soft sets fuzzy soft graphs operations on fuzzy soft graphs irregular fuzzy soft graphs decision making

1 Introduction

Molodtsov [24] initiated the concept of soft set theory as a new mathematical tool for dealing with uncertainties. Since then research on soft sets has been very active and received much attention from researchers worldwide. To extend the expressive power of soft sets, Jiang et al. [18] presented ontology-based soft sets, which extended soft sets with description logics. Ali et al. [10] proposed several new operations in soft set theory. Gong et al. [17] initiated the concept of bijective soft sets. Babitha and Sunil [11] extended the concepts of relations and functions in the context of soft set theory. At present, the combination of the soft set model and other mathematical models has received much attention. Maji et al. [20] introduced fuzzy soft sets, a more generalized notion combining fuzzy sets and soft sets. By using this definition of fuzzy soft sets, many interesting applications of fuzzy soft set theory have been expanded by some researchers. Roy and Maji [26] gave some applications of fuzzy soft sets. Ali et al. [9] discussed the fuzzy sets and fuzzy soft sets induced by soft sets. Som [28] defined soft relation and fuzzy soft relation on the theory of soft sets. The algebraic structure of soft set theory and fuzzy soft set theory dealing with uncertainties has also been studied in more detail. Aktas and Cagman [1] introduced a definition of soft groups, and derived their basic properties. In [15], Feng et al. defined the notion of a soft semiring. Zhan et al. [33, 34] discussed algebraic properties of rough soft hemirings and soft union sets to hemirings. Akram et al. [7] introduced the notion of fuzzy soft Lie algebras.

Fuzzy graph theory is finding an increasing number of applications in modeling real time systems where the level of information inherent in the system varies with different levels of precision. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional numerical models used in engineering and sciences and the symbolic models used in expert systems. Kaufmann’s initial definition of a fuzzy graph [19] was based on Zadeh’s fuzzy relations [32]. Rosenfeld [25] described the structure of fuzzy graphs obtaining analogs of several graph theoretical concepts. Bhattacharya [12] gave some remarks on fuzzy graphs and operations on fuzzy graphs were introduced by Mordeson and Nair in [22]. In [29], the definition of complement of a fuzzy graph was modified so that the complement of the complement is the original fuzzy graph, which agrees with the crisp graph case. Akram et al.[2 –6] introduced many new concepts, including bipolar fuzzy graphs, strong intuitionistic fuzzy graphs, intuitionistic fuzzy hypergraphs, soft graphs and fuzzy soft graphs. In this research article, we apply the concept of fuzzy soft sets to graphs. We present the concept of fuzzy soft graphs, various methods of their construction, and investigate some of their related properties. We discuss different types of irregular fuzzy soft graphs. We also describe an application of fuzzy soft graphs in social network. We have used standard definitions and terminologies in this paper. For other notations, terminologies and applications not mentioned in the paper, the readers are referred to [8 , 32,35].

2 Preliminaries

In this section, we review some basic concepts that are necessary for fully benefit of this paper.

Zadeh [31] initiated the notion of fuzzy subset of a set in 1965. A fuzzy setA on X is characterized by its membership function μ_A : X → [0, 1], where μ_A (x) is degree of membership of element x in fuzzy set A for each x ∈ X . A fuzzy relation on V is a fuzzy subset of V × V . A fuzzy relationν on V is a fuzzy relation on μ if ν (u, v) ≤ μ (u) ∧ μ (v) for all u, v in V. A fuzzy graph of a graph G^* = (V, E) is a pair G = (μ, ν), where μ and ν are fuzzy sets on V and V × V respectively, such that ν (uv) ≤ min(μ (u) , μ (v)) ∀ uv ∈ E . Note that ν (uv) =0 for all uv ∈ V × V - E. Soft set theory was proposed by Molodtsov in 1999. Let U be the universe of discourse and P be the universe of all possible parameters related to the objects in U. Each parameter is a word or a sentence. In most cases, parameters are considered to be attributes, characteristics or properties of objects in U. The pair (U, P) is also known as a soft universe. The power set of U is denoted by 𝓅(U).

Definition 2.1. [24] A pair 𝔖 = (φ, A) is called a soft set over U, where A ⊆ P is a parameter set and φ : A → 𝓅(U) is a set-valued mapping, called the approximate function of the soft set 𝔖. In other words, a soft set over U is a parameterized family of subsets of U. For any ε ∈ A, φ (ε) may be considered as set of ε-approximate elements of soft set (φ, A).

In other words, a soft set over U is a parameterized family of subsets of universe U. Thus a soft set over U can be written in a set of ordered pairs $(φ, A) = {(x, φ (x)) : x \in A, φ (x) \in 𝓅 (U)} .$ Maji et al. [20] combined two notions, namely, fuzzy set and soft set, and he defined fuzzy soft set in the following way:

Definition 2.2. [20] Let U be an initial universe, P the set of all parameters, A ⊂ P and $P (U)$ the collection of all fuzzy subsets of U. Then (F, A) is called fuzzy soft set, where $F : A \to P (U)$ is a mapping, called fuzzy approximate function of the fuzzy soft set (F, A).

Definition 2.3. [20] Let (F, A) and (K, B) be two fuzzy soft sets over U. We say that (F, A) is a fuzzy soft subset of (K, B) and write (F, A) Subset (G, B) if

A ⊆ B,

For any ɛ ∈ A, F (ɛ) ⊆ K (ɛ).

(F, A) and (K, B) are said to be fuzzy soft equal and write (F, A) = (K, B) if (F, A) Subset (K, B) and (K, B) Subset (F, A).

Definition 2.4. [20] A fuzzy soft set (F, A) is called null fuzzy soft set if F (ɛ) =0 for all ɛ ∈ A. A fuzzy soft set (F, A) is called absolute fuzzy soft set if F (ɛ) =1 for all ɛ ∈ A.

Definition 2.5. [20] Let (F, A) and (K, B) be two fuzzy soft sets over U.

The extended union of (F, A) and (K, B) is defined as the the fuzzy soft set (H, C) = (F, A) ⋁ _E (K, B) where (i) C = A ∪ B and (ii) for all ɛ ∈ C, $H (ɛ) = {\begin{matrix} F (ɛ), & if ɛ \in A ∖ B, \\ K (ɛ), & if ɛ \in B ∖ A, \\ F (ɛ) \lor K (ɛ), & if ɛ \in A \cap B . \end{matrix}$

The extended intersection of (F, A) and (K, B) is defined as the fuzzy soft set (H, C) = (F, A) ⋀ _E (K, B) where (i) C = A ∪ B and (ii) for all ɛ ∈ C, $H (ɛ) = {\begin{matrix} F (ɛ), & if ɛ \in A ∖ B, \\ K (ɛ), & if ɛ \in B ∖ A, \\ F (ɛ) \land K (ɛ), & if ɛ \in A \cap B . \end{matrix}$

Definition 2.6. [20] Let (F, A) be a fuzzy soft set over U . The complement of fuzzy set (F, A) is denoted by (F, A) ^c and defined by (F, A) ^c = (F^c, ¬ A), where, $F^{c} : \neg A \to P (U)$ such that F^c (ɛ) = (F (¬ ɛ)) ^c for all ɛ ∈ ¬ A.

Definition 2.7. [20] If (F, A) and (K, B) are two fuzzy soft sets then the fuzzy soft subset (R, C) of (F, A) × (K, B) is called a fuzzy soft relation, where C ⊂ A × B, for all (x, y) ∈ A × B, R (x, y) is a fuzzy subset of Q (x, y), and Q (x, y) = F (x) ∩ K (y).

3 Fuzzy soft graphs

Definition 3.1. A fuzzy soft graphG = (G^*, F, K, A) is a 4-tuple such that

G^* = (V, E) is a simple graph,

A is a nonempty set of parameters,

(F, A) is a fuzzy soft set over V,

(K, A) is a fuzzy soft set over E,

(F (e) , K (e)) is a fuzzy graph of G^* for all e ∈ A . That is, K (e) (xy) ≤ min {F (e) (x) , F (e) (y)} for all e ∈ A and x, y ∈ V. Note that K (e) (xy) =0 for all xy ∈ V × V - E and for all e ∈ A.

The fuzzy graph (F (e) , K (e)) is denoted by H (e) for convenience. In other words, a fuzzy soft graph is a parameterized family of fuzzy graphs of G^*. That is, G = (G^*, F, K, A) = {H (e) : e ∈ A}. The class of all fuzzy soft graphs of G^* is denoted by $F (G^{*}) .$

Example 3.2. Consider a simple graph G^* =(V, E) such that V = {a₁, a₂, a₃, a₄} and E ={a₁a₂, a₂a₃, a₃a₁, a₁a₄, a₂a₄, a₃a₄}. Let A = {e₁, e₂} be a parameter set and let (F, A) be afuzzy soft set over V with its fuzzy approximate function $F : A \to P (V)$ defined by

F (e₁) = {a₁|0.3, a₂|0.5, a₃|0.7, a₄|0.9}, F (e₂) = {a₁|0.7, a₂|0.3, a₃|0.4, a₄|0.6} .

Let (K, A) be a fuzzy soft set over E with its fuzzy approximate function $K : A \to P (E)$ defined by $\begin{matrix} K (e_{1}) = {a_{1} a_{2} | 0.2, a_{2} a_{3} | 0.4, a_{3} a_{1} | 0.3, a_{1} a_{4} | \\ 0.2, a_{2} a_{4} | 0.3, a_{3} a_{4} | 0.5}, \\ K (e_{2}) = {a_{1} a_{2} | 0.2, a_{2} a_{3} | 0.1, a_{3} a_{1} | 0.4, a_{1} a_{4} | \\ 0.0, a_{2} a_{4} | 0.2, a_{3} a_{4} | 0.3} . \end{matrix}$

Thus, fuzzy graphs corresponding to parameterse₁, e₂ are H (e₁) = (F (e₁) , K (e₁)), H (e₂) = (F (e₂) , K (e₂)) which are shown in Fig. 1.

Hence, G = {H (e₁) , H (e₂)} is a fuzzy soft graph of G^* . Tabular representation of a fuzzy soft graph is given in Table 1

Example 3.3. Consider a simple graph G^* =(V, E) such that V = {a₁, a₂, a₃} and E = {a₁a₂, a₂a₃, a₃a₁}. Let A = {e₁, e₂, e₃} be a parameter set. Let (F, A) be a fuzzy soft set over V corresponding to A which is represented by the Table 2.

Let (K, A) be a fuzzy soft set over E corresponding to A which is represented by the Table 3.

Thus, fuzzy graphs corresponding to parameters e₁, e₂ and e₃ are H (e₁) = (F (e₁) , K (e₁)), H (e₂) = (F (e₂) , K (e₂)) , H (e₃) = (F (e₃) , K (e₃)) as shown in Fig. 2.

Hence, G = {H (e₁) , H (e₂)} is a fuzzy soft graph over G^* .

Definition 3.4. Let G₁ = (F₁, K₁, A) and G₂ = (F₂, K₂, B) be two fuzzy soft graphs of G^*. Then G₂ is a fuzzy soft subgraph of G₁ if B ⊆ A, H₂ (e) is a partial fuzzy subgraph of H₁ (e) for all e ∈ B .

Example 3.5. Consider an undirected graph G^* = (V, E) such that V = {a₁, a₂, a₃, a₄, a₅, a₆} and E = {a₁a₂, a₂a₃, a₃a₄, a₄a₅, a₅a₆, a₆a₁}. Let A = {e₁, e₂, e₃} and B = {e₁, e₂} be parameter sets and let (F₁, A) be a fuzzy soft set over V with its approximate function $F_{1} : A \to P (V)$ defined by $\begin{matrix} F_{1} (e_{1}) = {a_{1} | 0.3, a_{2} | 0.1, a_{3} | 0.4, a_{4} | 0.6, a_{5} | 0.7, a_{6} | 0.4}, \\ F_{1} (e_{2}) = {a_{1} | 0.8, a_{2} | 0.4, a_{3} | 0.1, a_{4} | 0.3, a_{5} | 0.6, a_{6} | 0.5}, \\ F_{1} (e_{3}) = {a_{1} | 0.7, a_{2} | 0.2, a_{3} | 0.6, a_{4} | 0.5, a_{5} | 0.4, a_{6} | 0.3} . \end{matrix}$ Let (K₁, A) be a fuzzy soft set over E with its approximate function $K_{1} : A \to P (E)$ defined by $\begin{matrix} K_{1} (e_{1}) = {a_{1} a_{2} | 0.1, a_{2} a_{3} | 0.1, a_{3} a_{4} | 0.2, a_{4} a_{5} | 0.3, \\ a_{5} a_{6} | 0.4, a_{6} a_{1} | 0.2}, \\ K_{1} (e_{2}) = {a_{1} a_{2} | 0.4, a_{2} a_{3} | 0.1, a_{3} a_{4} | 0.1, a_{4} a_{5} | 0.2, \\ a_{5} a_{6} | 0.5, a_{6} a_{1} | 0.3}, \\ K_{1} (e_{3}) = {a_{1} a_{2} | 0.1, a_{2} a_{3} | 0.2, a_{3} a_{4} | 0.1, a_{4} a_{5} | 0.4, \\ a_{5} a_{6} | 0, a_{6} a_{1} | 0.2} . \end{matrix}$ Let (F₂, A) be a fuzzy soft set over V with its approximate function $F_{2} : B \to P (V)$ defined by $\begin{matrix} F_{2} (e_{1}) = {a_{1} | 0.2, a_{2} | 0.1, a_{3} | 0.2, a_{4} | 0.3, a_{5} | 0.4, a_{6} | 0.2}, \\ F_{2} (e_{2}) = {a_{1} | 0.6, a_{2} | 0.3, a_{3} | 0.1, a_{4} | 0.2, a_{5} | 0.3, a_{6} | 0.4} . \end{matrix}$ Let (K₂, A) be a fuzzy soft set over E with its approximate function $K_{2} : B \to P (E)$ given by $\begin{matrix} K_{2} (e_{1}) = {a_{1} a_{2} | 0.1, a_{2} a_{3} | 0.1, a_{3} a_{4} | 0.2, a_{4} a_{5} | 0.3, \\ a_{5} a_{6} | 0.2, a_{6} a_{1} | 0.1}, \\ K_{2} (e_{2}) = {a_{1} a_{2} | 0.3, a_{2} a_{3} | 0.1, a_{3} a_{4} | 0.0, a_{4} a_{5} | 0.1, \\ a_{5} a_{6} | 0.2, a_{6} a_{1} | 0.3} . \end{matrix}$

Hence $G_{1} = {H_{1} (e_{1}), H_{1} (e_{2}), H_{1} (e_{3})}, G_{2} = {H_{2} (e_{1}), H_{2} (e_{2})} \in F (G^{*}) .$ Clearly, B ⊆ A and H₂ (e) is a partial fuzzy subgraph of H₁ (e) for all e ∈ B . Hence G₂ is a fuzzy soft subgraph of G₁ .

Theorem 3.6. Let G₁ = (G^*, F₁, K₁, A) and G₂ = (G^*, F₂, K₂, B) be two fuzzy soft graphs. Then G₂ is a fuzzy soft subgraph of G₁ if and only if F₂ (e) ⊆ F₁ (e) and K₂ (e) ⊆ K₁ (e) for all e ∈ B.

Proof. Suppose that G₂ is a fuzzy soft subgraph of G₁ . Then B ⊆ A and H₂ (e) is a partial fuzzy subgraph of H₁ (e) for all e ∈ B . Clearly, F₂ (e) ⊆ F₁ (e) and K₂ (e) ⊆ K₁ (e) for all e ∈ B. Conversely, suppose that F₂ (e) ⊆ F₁ (e) and K₂ (e)⊆K₁ (e) for all e ∈ B. Since G₁ is a fuzzy soft graph of G^*, H₁ (e) is a fuzzy subgraph of G^* for all e ∈ A .

Since G₂ is a fuzzy soft graph of G^*, H₂ (e) is a fuzzy subgraph of G^* for all e ∈ B . Thus H₂ (e) is a partial fuzzy subgraph of H₁ (e) for all e ∈ B . Hence G₂ is a fuzzy soft subgraph of G₁ .□

Definition 3.7. The fuzzy soft graph H = (G^*, F₁, K₁, B) is called fuzzy soft subgraph of G = (G^*, F, K, A) induced by F₁ (a) (x) if

B ⊆ A,

F₁ (a) (x) ⊆ F (a) (x),

K₁ (a) (xy) = min {F₁ (a) (x) , F₁ (a) (y) , K (a) (xy)}

for all a ∈ A, x, y ∈V.

Definition 3.8. The fuzzy soft graph H = (G^*, F₁, K₁, B) is called spanning fuzzy soft subgraph of G = (G^*, F, K, A) if B ⊆ A, F₁ (a) (x) = F (a) (x), for all a ∈ A, x ∈ V. There is no any restriction for arc weights.

Definition 3.9. Let G₁ = (F₁, K₁, A) and G₂ =(F₂, K₂, B) be two fuzzy soft graphs of $G_{1}^{*} =$ (V₁, E₁) and $G_{2}^{*} = (V_{2}, E_{2})$ , respectively. The Cartesian product of of G₁ and G₂ is a fuzzy softgraph G = G₁⋉G₂ = (F, K, A × B), where (F, A×B) is a fuzzy soft set over V = V₁ × V₂, (K, A×B) is a fuzzy soft set over E = {((u, v₁) , (u, v₂)) : u ∈ V₁, (v₁, v₂) ∈ E₂} ∪ {((u₁, v) , (u₂, v)) : v ∈ V₂, (u₁, u₂) ∈ E₁}, and (F (a, b) , K (a, b)) is a fuzzy graph for all (a, b) ∈ A × B. That is,

F (a, b) (u, v) = F₁ (a) (u)∧F₂ (b) (v) , ∀ (u, v) ∈ V, (a, b) ∈ A × B,

K (a, b) ((u, v₁) , (u, v₂)) = F₁ (a) (u)∧K₂ (b) (v₁, v₂) , ∀ u ∈ V₁, (v₁, v₂) ∈ E₂,

K (a, b) ((u₁, v) , (u₂, v)) = F₂ (a) (v)∧K₁ (b) (u₁, u₂),∀ v ∈ V₂, (u₁, u₂) ∈ E₁ ∀ v ∈ V₂, (u₁, u₂) ∈ E₁.

H (a, b) = H₁ (a) ⋉H₂ (b) = {F₁ (a) × F₂ (b) , K₁(a) × K₂ (b)} for all (a, b) ∈ A × B is a fuzzy graph of G

Example 3.10. Let A = {e₁, e₂} and B = {e₁, e₃} be the sets of parameters. Consider the two fuzzy soft graphs G₁ = {H₁ (e₁) , H₁ (e₂)} and G₂ = {H₂ (e₁) , H₂ (e₃)} such that H₁ (e₁) = (F₁ (e₁) = {a₁|0.2, a₂|0.3, a₃|0.4}, K₁ (e₁) = {a₁a₂|0.1, a₁a₃|0.2}), H₁ (e₂) = (F₁ (e₂) = {a₁|0.5, a₂|0.7, b₃|0.6}, K₁ (e₂) = {a₁a₂|0.4, a₁a₃|0.3}), H₂ (e₁) = (F₂ (e₁) = {b₁|0.3, b₂|0.5, b₃|0.7, b₄|0.9}, K₂ (e₁) = {b₁b₂|0.2, b₂b₃|0.5, b₁b₄|0.1}), H₂ (e₃) = (F₂ (e₃) = ({b₁|0.7, b₂|0.4, b₃|0.6, b₄|0.8}, K₂ (e₃) = {b₁b₂|0.3, b₂b₃|0.2, b₁b₄|0.5}).

The Cartesian product of G₁ and G₂ is G₁⋉G₂= G = (H₃, A × B), where A × B = {(e₁, e₁) , (e₁, e₃) , (e₂, e₁) , (e₂, e₃)}, and H₃ (e₁, e₁) = H₁ (e₁) ⋉H₂ (e₁) , H₃ (e₁, e₃) = H₁ (e₁)⋉H₂ (e₃) , H₃ (e₂, e₁) = H₁ (e₂) ⋉H₂ (e₁) , H₃ (e₂, e₃)= H₁ (e₂) ⋉H₂ (e₃) are fuzzy graphs. H₃ (e₁, e₁)= H₁ (e₁) ⋉H₂ (e₁) is shown in Fig. 3.

In the similar way, other Cartesian productsH₃ (e₁, e₃) = H₁ (e₁) ⋉H₂ (e₃) , H₃ (e₂, e₁) = H₁ (e₂)⋉H₂ (e₁) , H₃ (e₂, e₃) = H₁ (e₂) ⋉H₂ (e₃) can bedrawn. Hence G = G₁⋉G₂ = {H₃ (e₁, e₁) , H₃ (e₁, e₃) , H₃ (e₂, e₁) , H₃ (e₂, e₃)} is a fuzzy soft graph.

Theorem 3.11. Cartesian product of two fuzzy soft graphs is a fuzzy soft graph.

Proof. Let G₁ and G₂ be two fuzzy soft graphs of $G_{1}^{*}$ and $G_{2}^{*} .$ Let (H₃, C) = {F (e_i) × F (e_j) , K (e_i) × K (e_j)} be the Cartesian product of G₁ and G₂, ∀ e_i ∈ A, ∀ e_j ∈ B for i = 1, 2, . . . , m, j = 1, 2, ⋯ , n . We claim that (H₃, C) is a fuzzy soft graph. Consider (K (e_i) × K (e_j)) ((a, b₁) (a, b₂) = min {F (e_i) (a) , K (e_j) (b₁, b₂)} for i = 1, 2, . . . , m, j = 1, 2, ⋯ , n .

min {F (e_i) (a) , min {F (e_j) (b₁) , F (e_j) (b₂)} for i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n .

min {F (e_i) (a) , F (e_j) (b₁)} , min {F (e_i) (a) , F (e_j)(b₂)} for i = 1, 2, . . . , m, j = 1, 2, ⋯ , n .

min {(F (e_i) × F (e_j)) (a, b₁) , (F (e_i) × F (e_j))(a, b₂)} for i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n .

Similarly, we can show that (K (e_i) × K (e_j)) ((a₁, b) (a₂, b) ≤ min {(F (e_i) × F (e_j)) (a₁, b) , (F (e_i) × F (e_j)) (a₂, b)} for i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n . Hence (H₃, C) is a fuzzy soft graph of G^* .□

Definition 3.12. The composition of G₁ and G₂ is a fuzzy soft graph G = G₁ ∘ G₂ = (F, K, A × B) , where (F, A × B) is a fuzzy soft set over V = V₁ × V₂, (K, A × B) is a fuzzy soft set over E = {((u, v₁) , (u, v₂)) : u ∈ V₁, (v₁, v₂) ∈ E₂} ∪ {((u₁, v) , (u₂, v)) : v ∈ V₂, (u₁, u₂) ∈ E₁} ∪ {((u₁, v₁) , (u₂, v₂)) : (u₁, u₂) ∈ E₁, v₁ ≠ v₂} and (F (a, b) , K (a, b)) is a fuzzy graph for all (a, b) ∈ A × B. That is,

F (a, b) (u, v) = F₁ (a) (u)∧F₂ (b) (v) , ∀ (u, v) ∈ V,

K (a, b) ((u, v₁) , (u, v₂)) = F₁ (a) (u)∧K₂ (b) (v₁, v₂), ∀ u ∈ V₁, (v₁, v₂) ∈ E₂,

K (a, b) ((u₁, v) , (u₂, v)) = F_{2_μ} (a) (v)∧K_{1_μ} (b) (u₁, u₂) , ∀ v ∈ V₂, (u₁, u₂) ∈ E₁,

K (a, b) ((u₁, v₁) , (u₂, v₂)) = K₁ (a) (u₁, u₂)∧F₂ (b) (v₁) ∧ F₂ (b) (v₂) , ∀ (u₁, u₂) ∈ E₁, where v₁ ≠ v₂.

H (a, b) = H₁ (a) ∘ H₂ (b) = {F₁ (a) ∘ F₂ (b) , K₁(a) ∘ K₂ (b)} for all (a, b) ∈ A × B is a fuzzy graph of G.

Example 3.13. Let A = {e₁, e₄} and B = {e₃} be the sets of parameters. Consider the two fuzzy soft graphs G₁ = {H₁ (e₁) , H₁ (e₄)} ={({a₁|0.6, a₂|0.9} , {a₁a₂|0.4}) , ({a₁|0.7, a₂|0.9} , {a₁a₂|0.5})} , G₂ ={H₂ (e₃)} = (b₁|0.7, b₂|0.8, b₃|0.9} , {b₁b₂|0.5, b₂b₃|0.7, b₃b₁|0.6}) . The composition of G₁ and G₂ isG₁ ∘ G₂ = G = (H₃, A × B), where A × B = {(e₁, e₃) , (e₄, e₃)} and H₃ (e₁, e₃) = H₁ (e₁) ∘ H₂ (e₃) , H₃ (e₄, e₃) = H₁ (e₄) ∘ H₂ (e₃) are fuzzy graphs.

H₁ (e₁) ∘ H₂ (e₃) is shown in Fig. 4. In the similar way, H₁ (e₄) ∘ H₂ (e₃) can be drawn. Hence G = G₁ ∘ G₂ = {H₃ (e₁, e₃) , H₃ (e₄, e₃)} is a fuzzy soft graph.

Theorem 3.14. Let G₁ and G₂ be two fuzzy soft graphs of $G_{1}^{*} = (V_{1}, E_{1})$ and $G_{2}^{*} = (V_{2}, E_{2}) .$ Then the composition (H₃, C) = {F (e_i) ∘ F (e_j) , K (e_i) ∘ K (e_j)} , i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n of G₁ and G₂ is a fuzzy soft graph.

Proof. As in the above Theorem 3.11, we prove thatK (e_i) ∘ K (e_j) ((a, b₁) (a, b₂) ≤ min {F (e_i) ∘ F (e_j)(a, b₁) , F (e_i) ∘ F (e_j) (a, b₂)} , K (e_i) ∘ K (e_j) ((a₁, c)(b₁, c) ≤ min {F (e_i) ∘ F (e_j) (a₁, b) , F (e_i) ∘ F (e_j)(a₂, b)} , for i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n . Now consider (a₁, b₁) (a₂, b₂) ∈ E^*, (a₁, a₂) ∈ E₁ and b₁ ≠ b₂ . Then $\begin{matrix} (K (e_{i}) \circ K (e_{j})) ((a_{1}, b_{1}) (a_{2}, b_{2})) \\ \leq min {F (e_{i}) (b_{1}), F (e_{j}) (b_{2}), K (e_{i}) (a_{1}, a_{2})}, \\ \leq min {F (e_{i}) (b_{1}), F (e_{j}) (b_{2}), \\ min {F (e_{i}) (a_{1}), F (e_{i}) (a_{2})}} \\ = min {min {F (e_{i}) (a_{1}), F (e_{j}) (b_{1})}, \\ min {F (e_{i}) (a_{2}), F (e_{j}) (b_{2})} \\ = min {(F (e_{i}) \circ F (e_{j})) (a_{1}, b_{1}), \\ (F (e_{i}) \circ F (e_{j})) (a_{2}, b_{2})}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix}$

Hence composition (H₃, C) = {F (e_i) ∘ F (e_j) , K (e_i) ∘ K (e_j)} , i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n of G₁ and G₂ is a fuzzy soft graph.□

Definition 3.15. Let $G_{1} = (G_{1}^{*}, F_{1}, K_{1}, A)$ and $G_{2} = (G_{2}^{*}, F_{2}, K_{2}, B)$ be two fuzzy soft graphs of $G_{1}^{*}$ and $G_{2}^{*}$ , respectively. Then union of G₁ and G₂, denoted by G₁⋓G₂, is a fuzzy soft graph (F, K, A ∪ B) , where (F, A ∪ B) is a fuzzy soft set over V = V₁ ∪ V₂, (K, A ∪ B) is a fuzzy soft set over E₁ ∪ E₂, and H (e) = (F (e) , K (e)) is a fuzzy graph for all e ∈ A ∪ B defined by $H (e) = {\begin{matrix} H_{1} (e), & if e \in A - B, \\ H_{2} (e), & if e \in B - A, \\ H_{1} (e) ⋓ H_{2} (e), & if e \in A \cap B, \end{matrix}$ where H₁ (e) ⋓H₂ (e) denotes the union of two fuzzy graphs of G for all e ∈ A ∩ B.

Example 3.16. Let $G_{1}^{*} = (V_{1}, E_{1}),$ where V₁ = {a₁, a₂, a₃} and E₁ = {a₁a₂, a₂a₃, a₃a₁} and $G_{2}^{*} = (V_{2}, E_{2}),$ where V₂ = {a₂, a₃, b₄, b₅} and E₂ = {a₂a₃, a₃b₅, b₄b₅} be two simple graphs. Let A = {e₁} and B = {e₁, e₂} be sets of parameters. Let (F₁, A) be a fuzzy soft set over V₁ with its approximate function $F_{1} : A \to P (V_{1})$ defined by F₁ (e₁) = {a₁|0.7, a₂|0.3, a₃|0.5} . Let (K₁, A) be a fuzzy soft set over E₁ with its fuzzy approximate function $K_{1} : A \to P (E_{1})$ defined by K₁ (e₁) = {a₁a₂|0.2, a₂a₃|0.3, a₃a₁|0.1} . Let (F₂, B) be a fuzzy soft set over V₂ with its approximate function $F_{2} : B \to P (V_{2})$ defined by $F_{2} (e_{1}) = {a_{2} | 0.6, a_{3} | 0.4, b_{4} | 0.5, b_{5} | 0.7},$

F₂ (e₂) = {a₂|0.7, a₃|0.2, b₄|0.1, b₅|0.4} . Let (K₂, B) be a fuzzy soft set over E₂ with its fuzzy approximate function $K_{2} : B \to P (E_{2})$ defined by K₂ (e₁) = {a₂a₃|0.4, a₃b₅|0.3, b₄b₅|0.5} , K₂ (e₂) ={a₂a₃|0.1, a₃b₅|0.2, b₄b₅|0.1} . Thus G₁ = {H₁ (e₁)} $\in F (G_{1}^{*})$ and $G_{2} = {H_{2} (e_{1}), H_{2} (e_{2})} \in F (G_{2}^{*}) .$ The union of G₁ and G₂ is (H₃, C) , where C = {e₁, e₂}, H₃ (e₁) = H₁ (e₁) ⋓H₂ (e₁) = $\begin{matrix} ({a_{1} | 0.7, a_{2} | 0.3, a_{3} | 0.5, b_{4} | 0.4, b_{5} | 0.7}, {a_{1} a_{2} | 0.2, \\ a_{2} a_{3} | 0.3, a_{3} a_{1} | 0.4, a_{3} b_{5} | 0.2, b_{4} b_{5} | 0.4}), \end{matrix}$

H₃ (e₂) = H₁ (e₂) ⋓H₂ (e₂ = H₂ (e₂) are fuzzy graphs.

H₃ (e₁) = H₁ (e₁) ⋓H₂ (e₁) is shown in Fig. 4. In the similar way, H₃ (e₂) can be drawn. Hence G = G₁⋓G₂ = {H₃ (e₁) , H₃ (e₂)} is a fuzzy soft graph.

Theorem 3.17. Let G₁ and G₂ be two fuzzy soft graphs over G^* such that A ∩ B ≠ ∅ . Then their union G₁⋓G₂ is a fuzzy soft graph over G^* .

Proof. The union of G₁ and G₂ is defined by G₁⋓G₂ = (H₃, C), where C = A ∪ B and $H_{3} (e) = {\begin{matrix} H_{1} (e), & if e \in A - B, \\ H_{2} (e), & if e \in B - A, \\ H_{1} (e) ⋓ H_{2} (e), & if e \in A \cap B . \end{matrix}$

Since $G_{1} \in F (G_{1}^{*})$ then H₁ (e) is a fuzzy graph for all e ∈ A . Since $G_{2} \in F (G_{2}^{*})$ then H₂ (e) is a fuzzy graph for all e ∈ B . Since union of two fuzzy graphs is a fuzzy graph, H₁ (e) ⋓H₂ (e) is a fuzzy graph for all e ∈ A ∩ B . Therefore, H₃ (e) is a fuzzy graph for all e ∈ C . Thus (H₃, C) is a fuzzy soft graph over G^* .□

Definition 3.18. Let $G_{1} = (G_{1}^{*}, F_{1}, K_{1}, A)$ and $G_{2} = (G_{2}^{*}, F_{2}, K_{2}, B)$ be two fuzzy soft graphs of $G_{1}^{*}$ and $G_{2}^{*}$ , respectively. Then join of G₁ and G₂, denoted by G₁ + G₂, is a fuzzy soft graph (F, K, A ∪ B) , where (F, A ∪ B) is a fuzzy soft set over V = V₁ ∪ V₂, (K, A ∪ B) is a fuzzy soft set over $E_{1} \cup E_{2} \cup \overset{´}{E}$ , where $\overset{´}{E}$ is the set of all edges joining the nodes of V₁ and V₂. In this construction it is assumed that V₁∩ V₂ ≠ ∅, and H (e) = (F (e) , K (e)) is a fuzzy graph for all e ∈ A ∪ B defined by $H (e) = {\begin{matrix} H_{1} (e), & if e \in A - B, \\ H_{2} (e), & if e \in B - A, \\ H_{1} (e) + H_{2} (e), & if e \in A \cap B, \end{matrix}$ where H₁ (e) + H₂ (e) denotes the join of two fuzzy graphs of G for all e ∈ A ∩ B.

Example 3.19. Let A = {e₁, e₂} and B = {e₁, e₃} be parameters sets. Let G₁ and G₂ be two fuzzy soft graphs as follows: G₁ = {H₁ (e₁) , H₁ (e₂)}, where H₁ (e₁) = ({a₁|0.5, a₂|0.4, a₃|0.2} , {a₁, a₂|0.3, a₁a₃|0.1}) , H₁ (e₂) =({a₁|0.7, a₂|0.8, a₃|0.9} , {a₁a₂|0.6, a₁a₃|0.6}) . G₂= {H₂ (e₁) , H₂ (e₃)} , where $\begin{matrix} H_{2} (e_{1}) = ({b_{1} | 0.3, b_{2} | 0.6, b_{3} | 0.4, b_{4} | 0.2}, \\ {b_{1} b_{2} | 0.2, b_{2} b_{3} | 0.4, b_{1} b_{4} | 0.2}), \\ H_{2} (e_{3}) = ({b_{1} | 0.4, b_{2} | 0.9, b_{3} | 0.7, b_{4} | 0.9}, \\ {b_{1} b_{2} | 0.4, b_{2} b_{3} | 0.2, b_{1} b_{4} | 0.3}) . \end{matrix}$

Join of G₁ and G₂ is G₁ + G₂ = (H, C), where C = A ∪ B = {e₁, e₂, e₃} and, H (e₁) = H₁ (e₁) + H₂ (e₁) , H (e₂) = H₁ (e₂) , H (e₃) = H₂ (e₃) ,

H (e₁) = H₁ (e₁) + H₂ (e₁) is shown in Fig. 7. In the similar way, other join fuzzy graphs can be drawn.

Hence G = G₁ + G₂ = {H (e₁) , H (e₂) , H (e₃)} is a fuzzy soft graph.

Remark. Join of two fuzzy soft graphs is a fuzzy soft graph.

Proposition 3.20. If G₁ and G₂ are two strong fuzzy soft graphs then their join G₁ + G₂ is also a strong fuzzy soft graph.

Definition 3.21. Let G₁ and G₂ be two fuzzy soft graphs of $G_{1}^{*}$ and $G_{2}^{*} .$ The intersection of G₁ and G₂ is denoted by G₁CapG₂ and is defined by G₁CapG₂ = (H₃, C), where C = A ∪ B and $H_{3} (e) = {\begin{matrix} H_{1} (e), & if e \in A - B, \\ H_{2} (e), & if e \in B - A, \\ H_{1} (e) \cap H_{2} (e), & if e \in A \cap B . \end{matrix}$ where H₁ (e) ∩ H₂ (e) denotes the intersection of two fuzzy graphs.

Example 3.22. Consider the simple graphs $G_{1}^{*} =$ (V₁, E₁), where V₁ = {a₁, a₂, a₃, a₄} and E₁ ={a₁a₂, a₂a₄, a₃a₄} and $G_{2}^{*} = (V_{2}, E_{2})$ , where V₂ ={a₂, a₃, a₄, b₁} and E₂ = {b₂b₃, b₃b₄, b₄b₁, a₂b₁, a₂a₄, a₃b₁} . Let A = {e₃, e₄} and B = {e₃} be parameter set. Let (F₁, A) be a fuzzy soft set over V₁ with its approximate function $F_{1} : A \to P (V_{1})$ defined by F₁ (e₃) = {a₁|0.5, a₂|0.6, a₃|0.4, a₄|0.3} , F₁ (e₄) = {a₁|0.7, a₂|0.5, a₃|0.9, a₄|0.1} .

Let (K₁, A) be a fuzzy soft set over E₁ with its fuzzy approximate function $K_{1} : A \to P (E_{1})$ defined by K₁ (e₃) = {a₁a₂|0.4, a₂a₃|0.2, a₃a₄|0.1} , K₁ (e₄) = {a₁a₂|0.3, a₂a₃|0.1, a₃a₄|0.1} .

Hence G₁ = {H₁ (e₃) , H₁ (e₄)} is a fuzzy soft graph of $G_{1}^{*} .$ Let (F₂, B) be a fuzzy soft set over V₂ with its approximate function $F_{2} : B \to P (V_{2})$ defined by F₂ (e₁) = {a₂|0.5, a₃|0.6, a₄|0.4, b₁|0.3} . Let (K₂, B) be a fuzzy soft set over E₂ with its fuzzy approximate function $K_{2} : B \to P (E_{2})$ defined by $\begin{matrix} K_{2} (e_{1}) = {a_{2} a_{3} | 0.2, a_{3} a_{4} | 0.1, a_{4} b_{1} | 0.2, a_{2} b_{1} | 0.3, \\ a_{2} a_{4} | 0.1, a_{3} b_{1} | 0.2} . \end{matrix}$

Hence G₂ = {H₂ (e₃)} is a fuzzy soft graph of $G_{2}^{*} .$ The intersection of G₁ and G₂ is (H₃, C), where C = A ∪ B = {e₃, e₄} and, H₃ (e₃) = H₁ (e₃) ∩ H₂ (e₃) = ({a₂|0.5, a₃|0.4, a₄|0.3} , {a₂a₃|0.2, a₃a₄|0.1}), H₃ (e₄) = H₁ (e₄) .

Theorem 3.23. Let G₁ and G₂ be two fuzzy soft graphs of G^* such that A ∩ B = ∅ . Then intersection G₁CapG₂ is a fuzzy soft graph over G^* .

Definition 3.24. Let $G_{1} = (G_{1}^{*}, F_{1}, K_{1}, A)$ and $G_{2} = (G_{2}^{*}, F_{2}, K_{2}, B)$ be two fuzzy soft graphs of $G_{1}^{*}$ and $G_{2}^{*}$ , respectively. Then cross product of G₁ and G₂ is a fuzzy soft graph G = G₁circledcircG₂ = (F, K, A × B) , where (F, A × B) is a fuzzy soft set over V = V₁ × V₂, (K, A × B) is a fuzzy soft set over E = {((u₁, v₁) , (u₂, v₂)) : (u₁, u₂) ∈ E₁, (v₁, v₂) ∈ E₂}, and (F (a, b) , K (a, b)) is a fuzzy graph for all (a, b) ∈ A × B. That is,

F (a, b) (u, v) = F₁ (a) (u)∧F₂ (b) (v) , ∀ (u, v) ∈ V, (a, b) ∈ A × B,

K (a, b) ((u₁, v₁) , (u₂, v₂)) = K_{1_μ} (a) (u₁, u₂) ∧ K_{2_μ} (b) (v₁, v₂) , ∀ (u₁, u₂) ∈ E₁, (v₁, v₂) ∈ E₂.

H (a, b) = H₁ (a) circledcircH₂ (b) = {F₁ (a) circledcircF₂ (b) , K₁(a) circledcircK₂ (b)} for all (a, b) ∈ A × B is a fuzzy graph.

Definition 3.25. Let $G_{1} = (G_{1}^{*}, F_{1}, K_{1}, A)$ and $G_{2} = (G_{2}^{*}, F_{2}, K_{2}, B)$ be two fuzzy soft graphs of $G_{1}^{*}$ and $G_{2}^{*}$ , respectively. Then lexicographic product of G₁ and G₂ is a fuzzy soft graph G = G₁ ⊙ G₂ = (F, K, A × B) , where (F, A × B) is a fuzzy soft set over V = V₁ × V₂, (K, A × B) is a fuzzy soft set over E = {((u, v₁) , (u, v₂)) : u ∈ V₁, (v₁, v₂) ∈ E₂} ∪ {((u₁, v₁) , (u₂, v₂)) : (u₁, u₂) ∈ E₁, (v₁, v₂) ∈ E₂}, and (F (a, b) , K (a, b)) is a fuzzy graph for all (a, b) ∈ A × B. That is,

F (a, b) (u, v) = F₁ (a) (u)∧F₂ (b) (v) , ∀ (u, v) ∈ V, (a, b) ∈ A × B,

K (a, b) ((u, v₁) , (u, v₂)) = F₁ (a) (u)∧K₂ (b) (v₁, v₂) , ∀ u ∈ V₁, (v₁, v₂) ∈ E₂,

K (a, b) ((u₁, v₁) , (u₂, v₂)) = K₁ (a) (u₁, u₂) ∧ K₂ (b) (v₁, v₂) , ∀ (u₁, u₂) ∈ E₁, (v₁, v₂) ∈ E₂,

H (a, b) = H₁ (a) ⊙ H₂ (b) for all (a, b) ∈ A × B is a fuzzy graph of G.

Definition 3.26. Let $G_{1} = (G_{1}^{*}, F_{1}, K_{1}, A)$ and $G_{2} = (G_{2}^{*}, F_{2}, K_{2}, B)$ be two fuzzy soft graphs of $G_{1}^{*}$ and $G_{2}^{*}$ , respectively. Then strong product of of G₁ and G₂ is a fuzzy soft graph G = G₁ ⊗ G₂ = (F, K, A × B) , where (F, A × B) is a fuzzy soft set over V = V₁ × V₂, (K, A × B) is a fuzzy soft set over E = {((u, v₁) , (u, v₂)) : u ∈ V₁, (v₁, v₂) ∈ E₂} ∪ {((u₁, v) , (u₂, v)) : v ∈ V₂, (u₁, u₂) ∈ E₁} ∪ {((u₁, v₁) , (u₂, v₂)) : (u₁, u₂) ∈ E₁, (v₁, v₂) ∈ E₂}, and (F (a, b) , K (a, b)) is a fuzzy graph for all (a, b) ∈ A × B. That is,

F (a, b) (u, v) = F₁ (a) (u)∧F₂ (b) (v) , ∀ (u, v) ∈ V,

K (a, b) ((u, v₁) , (u, v₂)) = F₁ (a) (u)∧K₂ (b) (v₁, v₂) , ∀ u ∈ V₁, (v₁, v₂) ∈ E₂,

K (a, b) ((u₁, v) , (u₂, v)) = F₂ (a) (v) ∧ K₁ (b) (u₁, u₂) , ∀ v ∈ V₂, (u₁, u₂) ∈ E₁,

K (a, b) ((u₁, v₁) , (u₂, v₂)) = K₁ (a) (u₁, u₂) ∧ K₂ (b) (v₁, v₂) , ∀ (u₁, u₂) ∈ E₁, (v₁, v₂) ∈ E₂.

H (a, b) = H₁ (a) ⊗ H₂ (b) for all (a, b) ∈ A × B is a fuzzy graph of G.

We state the following proposition without its proof.

Proposition 3.27. If G₁ and G₂ are fuzzy soft graphs, then G₁circledcircG₂, G₁ ⊙ G₂, and G₁ ⊗ G₂ are fuzzy soft graphs.

Definition 3.28. The complement of a fuzzy soft graph G = (F, K, A) of G^* is denoted by G^c and is defined by G^c = (F^c, K^c, A^c) where A^c = A, F^c (e) = F (e) and K^c (a, b) = F (a) ∧ F (b) - K (a, b) for all a, b ∈ V .

In other words, the complement of a fuzzy soft graph G is the complement of fuzzy graph H (e) for all e ∈ A .

Example 3.29. Consider an undirected graph G^* = (V, E), where V = {a₁, a₂, a₃, a₄} and $E = {a_{1} a_{2}, a_{2} a_{4}, a_{3} a_{4}} .$

Let A = {e₁, e₂} and (F, A) be a fuzzy soft set over V with its approximate function $F : A \to P (V)$ given by F (e₁) = {a₁|0.3, a₂|0.7, a₃|0.2, a₄|0.4} , F (e₂) = {a₁|0.5, a₂|0.9, a₃|0.7, a₄|0.6} . Let (K, A)be a fuzzy soft set over E with its approxi-mate function $K : A \to P (E)$ given by K (e₁) ={a₁a₂|0.2, a₂a₄|0.3, a₃a₄|0.1} , K (e₂) = {a₁a₂|0.4, a₂a₄|0.6, a₃a₄|0.2} . By routine calculations, it is easy to see that H (e₁) and H (e₂) are fuzzy graphs of G^* shown in Fig. 7.

Hence, G is a fuzzy soft graph. Now the complement of fuzzy soft graph is the complement of fuzzy graphs H (e₁) and H (e₂) which are shown in Fig. 8.

Definition 3.30. A fuzzy soft graph G is self- complementary if G ≈ G^c

Example 3.31. Consider a fuzzy soft graph G = {H (e₁) , H (e₂)} , where H (e₁) = ({a₁|0.2, a₂|0.5, a₃|0.2, a₄|0.4}, $\begin{matrix} {a_{1} a_{2} | 0.1, a_{2} a_{3} | 0.1, a_{3} a_{4} | 0.1, a_{4} a_{1} | \\ 0.1, a_{1} a_{3} | 0.1, a_{2} a_{4} | 0.2}) \end{matrix}$ H (e₂) = ({a₁|0.4, a₂|0.6, a₃|0.4, a₄|0.7}, $\begin{matrix} {a_{1} a_{2} | 0.2, a_{2} a_{3} | 0.2, a_{3} a_{4} | 0.2, a_{4} a_{1} | 0.2, \\ a_{1} a_{3} | 0.2, a_{2} a_{4} | 0.3}) \end{matrix}$ are shown in Fig. 9. By routine calculations, it is easy to see that G is self- complementary.

Proposition 3.32. If G is a fuzzy soft graph over G^* then G^c is also a fuzzy soft graph.

Proposition 3.33.If G is a strong fuzzy soft graph over G^*, then G^c is also a strong fuzzy soft graph.

Proof. Since G is a strong fuzzy soft graph over G^*, then $K (e) (ab) = min {F (e) (a), F (e) (b)}$ (1) for all xy ∈ E, e ∈ A .

By definition of complement, we have K^c (e)(ab) = min {F (e) (a) , F (e) (b)} - K (e) (ab) for all ab ∈ E, e ∈ A . If ab ∈ E, then K^c (e) (ab) = min {F (e) (a) , F (e) (b)} - min {F (e) (a) , F (e) (b)} =0, using (1). If ab ∉ E, then K^c (e) (ab) = min {F (e)(a) , F (e) (b)} -0 = min {F (e) (a) , F (e) (b)} for all ab ∈ E, e ∈ A . Hence G^c is a strong fuzzy soft graph.□

Theorem 3.34. If G and G^c are strong fuzzy soft graphs of G^*. Then G ∪ G^c is a complete fuzzy soft graph.

We now discuss different types of irregular fuzzy soft graphs.

Definition 3.35. A fuzzy soft graph G = (G^*, F, K, A) is called an irregular fuzzy soft graph if H (e) = (F (e) , K (e)) is irregular fuzzy graph for all e ∈ A . Equivalently, a fuzzy soft graph G is called an irregular fuzzy soft graph if there is a vertex which is adjacent to the vertices with distinct degrees in H (e) for all e ∈ A.

Definition 3.36. A fuzzy soft graph G = (G^*, F, K, A) is called a neighbourly irregular fuzzy soft graph if H (e) = (F (e) , K (e)) is neighbourly irregular fuzzy graph for all e ∈ A . Equivalently, a fuzzy soft graph G is called a neighbourly irregular fuzzy soft graph if every two adjacent vertices have distinct degrees in H (e) for all e ∈ A.

Definition 3.37. A fuzzy soft graph G = (G^*, F, K, A) is called a highly irregular fuzzy soft graph if H (e) = (F (e) , K (e)) is highly irregular fuzzy graph for all e ∈ A . Equivalently, a fuzzy soft graph G is called a highly irregular fuzzy soft graph if every vertex is adjacent to the vertices of distinct degrees in H (e) for all e ∈ A.

Remark. A highly irregular fuzzy soft graph may not be a neighbourly irregular fuzzy soft graph.

Remark. A neighbourly irregular fuzzy soft graph may not be a highly irregular fuzzy soft graph.

We state the following theorem without proof which represent the relationship between neighbourly and highly irregular fuzzy soft graph.

Theorem 3.38. A fuzzy soft graph G is both a neighbourly irregular and highly irregular fuzzy soft graph if and only if the degrees of all the vertices are distinct.

Definition 3.39. A fuzzy soft graph G = (G^*, F, K, A) is called a totally irregular fuzzy soft graph if H (e) = (F (e) , K (e)) is a totally irregular fuzzy graph for all e ∈ A .

Definition 3.40. A fuzzy soft graph G = (G^*, F, K, A) is called a neighbourly totally irregular fuzzy soft graph if H (e) = (F (e) , K (e)) is neighbourly totally irregular fuzzy graph for all e ∈ A .

Theorem 3.41. If G is neighbourly irregular fuzzy soft graph then G^c is not a neighbourly irregular fuzzy soft graph.

Theorem 3.42. Let G be a fuzzy soft graph. If G is a neighbourly irregular fuzzy soft graph and F is a constant function then G is a neighbourly totally irregular fuzzy soft graph.

Proof. Suppose that G is a neighbourly irregular fuzzy soft graph. Then all the adjacent vertices have distinct degrees. Let a₁ and a₂ be two adjacent vertices with distinct degrees s_i and t_i respectively in H (e_i) for i = 1, 2, . . . , n. That is, deg(a₁) = s_i and deg(a₂) = t_i in $\tilde{H} (e_{i})$ , where s_i ≠ t_i for i = 1, 2, . . . , n .

Since F is a constant function, therefore F (e_i) (a₁) = c_i = F (e_i) (a₂), where c_i is a constant function and c_i ∈ [0, 1] for all e_i ∈ A for i = 1, 2, 3, . . . , n . Now tdeg (a₁) = deg(a₁) + c_i = s_i + c_i, tdeg (a₂) = deg(a₂) + c_i = t_i + c_i in fuzzy subgraph H (e_i) for all e_i ∈ A for i = 1, 2, . . . , n . Claim: tdeg (a₁) ≠ tdeg (a₂) .

On contrary, suppose that tdeg (a₁) = tdeg (a₂) .

Then s_i + c_i = t_i + c_i s_i - t_i = c_i - c_i = 0 ⇒ s_i = t_i, which is a contradiction to the fact that s_i ≠ t_i . ⇒ tdeg (a₁) ≠ tdeg (a₂) in H (e_i), i = 1, 2, . . . , n . ⇒ no two adjacent vertices have same total degrees in H (e_i) for i = 1, 2, . . . , n . ⇒ H (e_i) is neighbourly irregular fuzzy graph for i = 1, 2, . . . , n . Hence G is a neighbourly totally irregular fuzzy soft graph.□

Theorem 3.43. Let G be a fuzzy soft graph. If G is a neighbourly totally irregular fuzzy soft graph and F is a constant function then G is a neighbourly irregular fuzzy soft graph.

Proof. Suppose that G is a neighbourly totally irregular fuzzy soft graph. Then all the adjacent vertices have distinct total degrees. Let a₁ and a₂ be two adjacent vertices with distinct total degrees m_i and n_i respectively in H (e_i) for i = 1, 2, . . . , n. That is, tdeg (a₁) = m_i and tdeg (a₂) = n_i in H (e_i), where m_i ≠ n_i for i = 1, 2, . . . , n .

Since F is a constant function, therefore F (e_i)(a₁) = f_i = F (e_i) (a₂), where f_i is a constant function, f_i ∈ [0, 1] for all e_i ∈ A for i = 1, 2, 3, . . . , n . Now deg (a₁) = tdeg (a₁) - f_i = m_i - f_i, deg (a₂) = tdeg (a₂) - f_i = n_i - f_i

Claim: deg (a₁) ≠ deg (a₂) .

On contrary, suppose that deg (a₁) = deg (a₂) .

Then m_i - f_i = n_i - f_i m_i - n_i = f_i - f_i = 0 ⇒ m_i = n_i, which is a contradiction to the fact that m_i ≠ n_i . ⇒ deg (a₁) ≠ deg (a₂) in H (e_i) for i = 1, 2, . . . , n . ⇒ no two adjacent vertices have same degrees. ⇒ H (e_i) is neighbourly irregular fuzzy graph for i = 1, 2, . . . , n . Hence G is a neighbourly irregular fuzzy soft graph.□

Proposition 3.44. Let G be a fuzzy soft graph. If G is both neighbourly irregular and neighbourly totally irregular fuzzy soft graph, then F need not be a constant function.

Remark. If G₁ is neighbourly irregular fuzzy soft graph, then fuzzy soft subgraph G₂ of G₁ may not be a neighbourly irregular fuzzy soft graph.

Remark. A complete fuzzy soft graph may not be a neighbourly irregular fuzzy soft graph.

Proposition 3.45. A neighbourly irregular fuzzy soft graph may not be a neighbourly totally irregular fuzzy soft graph.

Proposition 3.46. A totally neighbourly irregular fuzzy soft graph may not be a neighbourly irregular fuzzy soft graph.

Example 3.47. We consider the crisp graph G^* = (V, E) where V = {a₁, a₂, a₃, a₄, a₅, a₆} and E = {a₁a₂, a₂a₃, a₃a₄, a₄a₅, a₅a₆, a₆a₁, a₃a₆}. Let A = {e₁, e₂}. Consider the fuzzy subgraphs H (e₁) and H (e₂) given below:

H (e₁) = ({a₁|0.5, a₂|0.8, a₃|0.4, a₄|0.3, a₅|0.4, a₆|0.1} , {a₁a₂|0.3, a₂a₃|0.2, a₃a₄|0.1, a₄a₅|0.2, a₅a₆|0.1, a₆a₁|0.1, a₃a₆|0.1}) , H (e₂) = ({a₁|0.3, a₂|0.5, a₃|0.7, a₄|0.9, 5|0.5, a₆|0.2} , {a₁a₂|0.3, a₂a₃|0.4, a₃a₄|0.5, a₄a₅|0.5, a₅a₆|0.1, a₆a₁|0.2, a₃a₆|0.1}) . All the adjacent vertices have distinct total degrees so H (e₁) and H (e₂) are neighbourly totally irregular fuzzy graphs. Hence G is neighbourly totally irregular fuzzy soft graph. But deg (a₄) = deg (a₅) = deg (a₆) =0.3 in H (e₁) and deg (a₃) = deg (a₄) =1 in H (e₂) . Therefore H (e₁) and H (e₂) are not neighbourly irregular fuzzy graphs. Hence G is not neighbourly irregular fuzzy soft graph.

4 Applications of fuzzy soft graphs

In this section, we describe applications of fuzzy soft graphs in social network and road network. with complex decision problems.

1. Application of fuzzy soft graphs in social network

Many practical problems can be represented by graphs. In studies of behavior of group persons, it is noticed that certain persons have influence thinking of others. An influence graph is a directed graph which can be used to model this behavior. In fuzzy influence graph, if vertices represent the persons and its membership degree represent the authority of persons and edges represent the influence of a person on another person in the social network, then we can find the most influential person within the social network. Now we discuss a fuzzy soft model to find out the most influential person in the department of industry with respect to different attributes of employees, including conflict, cooperative, industrious, performance.

We consider a department of the industry having employees and their designations as shown in Table 4.

A survey conducted on department of industry produce the following results:

Bashir is such a person of an industry who is active in every crucial decision and responds calmly in stressful situations.

Bashir and Raheem have a good relationship and have worked together. Raheem values Bashir.

Bashir is a right hand of the Chief Engineer. Like Raheem, the Chief Engineer also values Bashir.

Nazeer has a great influence in the development team. He has provided directions to Community Development Agency in coordinating redevelopment activities.

Kareem’s ability to communicate effectively and politely enables him to perform his job effectively. Also, he has good-natured and pleasant to others.

Consider the directed graph with vertex set =V={Saleem, Bashir, Raheem, Kareem, Imran, Nazeer, Nasir, Kashif }. The vertices represent employees and directed edges represent any relationship between them. Let set of attributes =A={e₁ = cooperation, e₂ = conflict }. A fuzzy soft graph G = {H (e) = (F (e) , K (e)) : e ∈ A} is represented by the following Table 4.

A fuzzy influence graph corresponding to employee’s cooperation, H (e₁) = (F (e₁) , K (e₁)), is shown in Fig. 10.

The vertices represent the employees and the membership degrees of the vertices represent the power of cooperation of employees in the industry. For example, Raheem has hold 0.6, that is, 60% power of cooperation within the industry. The edges represent dependence in their cooperation on one another. If there is no edge between any two employees, it means that two employees do not depend on each other. The degree of membership can be interpreted as a percentage of dependence. We can easily seen in Fig. 10 that cooperation power of Saleem, Bashir and Nazeer are more than all other employees. If we talk about Nazeer, he depends on Bashir, Nasir and Imran. Their power of cooperation in the industry is 30% , 90% and 50% respectively. The employees that depend on Nazeer are Kareem and Kashif and their power of cooperation is 50% and 60% respectively. While Saleem, Raheem, Imran and Nazeer depend on Bashir. Their power of cooperation in the industry is 90% , 70% , 60% and 50% respectively. So it is vivid that Bashir is the more important and cooperative person in the industry because many people depend on him.

A fuzzy influence graph corresponding to attribute conflict, H₂ (e₂) = (F (e₂) , K (e₂)), is shown in Fig. 11 and tabular representation of this graph is given in Table 5.

The vertices represent the employees and membership degree of vertices (Table 4) represent their goodness (behavior). The edges represent conflict between two employees. If there is no edge between any two employees, it means that the employees have no conflict. We can seen in Fig. 11 that Saleem and Kareem have more membership degree of their goodness than all other employees. Raheem and Imran have more conflicts in the industry. Raheem is 50% well-behaved but his conflicts are with Nazeer, Nasir and Kashif which are 60% , 60% and 70% well-behaved, respectively. Now if we talk about Imran, he is 50% well-behaved but his conflicts are with Saleem, Bashir and Kareem which are 90% 70% and 80% well-behaved in the industry respectively. So Imran have conflicts with those employees which are more well-behaved in the industry. Hence, Imran is more combative person in the industry.

2. Application of fuzzy soft graphs in road network

We now discuss a fuzzy soft graph model for road network. We take area of any city with intersections and roads. Consider set of vertices= V = {Abdullahpur(Ab), Jaal(Jl), Saleemi chowk(Sc), Mashali form(Mf), Kohinoor chowk(Kc), Tazab mil(Tm), Akhari stop(As), Kashmir pul(Kp), Madina town(Mt), Khuram chowk(Khc)} and the set of attributes = A = {e₁=day, e₂= night }. A fuzzy soft graph G = {H (e) = (F (e) , K (e)) : e ∈ A} is represented by the following Table 6.

We can see the flow of traffic in our road network in day time and night time. The fuzzy digraph corresponding to parameter day e₁, H (e₁) = (F (e₁) , K (e₁)), as shown in Fig. 12.

Fuzzy digraph corresponding to parameter night e₂, H (e₂), is shown in Fig. 13.

In fuzzy road network models, vertices represent intersections and its membership degree represent percentage of traffic on intersections and edges represent roads and its membership degree represent percentage of traffic on roads. Thus a fuzzy soft graph tells us about flow of traffic on intersections and roads with respect to different parameters.

5 Conclusions and future work

There has been a rapid growth of interest in developing soft computing models that are efficient to deal with vagueness and uncertainty. For this purpose, a fuzzy soft set theory has been developed to handle vagueness and indeterminacy in applications including decision making problems. In this article, we have applied these soft computing models in combination to study vagueness and uncertainty in graphs. We have defined some operations on fuzzy soft graph and investigated some of their properties. We have also discussed applications of fuzzy soft graphs. We are extending our research of fuzzification to (1) Interval-valued fuzzy soft graphs; (2) Bipolar fuzzy soft hypergraphs, (3) Fuzzy rough soft graphs, and (4) Intuitionistic fuzzy soft graphs.

Footnotes

Acknowledgments

The authors are highly thankful to an Associate Editor and the referees for their valuable comments and suggestions for improving the quality of paper.

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