Hesitant fuzzy graphs and their applications in decision making

Abstract

In this study, hesitant fuzzy graph (HFG) structure and some concepts related to HFGs such as homomorphism, isomorphism, weak isomorphism and co-weak isomorphism are introduced. Also, operations of Cartesian product, direct product, lexicographical product and strong product between two HFGs are defined, and examples are given related these operations. Finally, a decision making algorithm and an application of HFGs in decision making are given.

Keywords

Hesitant fuzzy set hesitant fuzzy graph product operations homomorphism isomorphism decision making

1. Introduction

The concept of fuzzy set, which is a useful tool to model vague and indeterminant information in real world, was defined by Zadeh [54] in 1965. A fuzzy set is characterized by its membership function. In a fuzzy set, each element in reference set has a membership degree, and there isn’t any hesitation on this membership degree. This case isn’t convenience to model some problems in the real life. Therefore, in 1986 Atanassov proposed the concept of intuitionistic fuzzy set which permits us to include some hesitations on membership degree. In classical set theory, we know that there are two important properties, one of them is that in a set ordering of elements doesn’t matter and the other is that repeated elements in a set are redundant. But in some cases, we may need a structure containing repeated elements. To express these situations Knuth [24] defined the concept of multi-sets(or bags). In 1986, Yager discussed operations of bags such as union, intersection and addition. He also gave a definition of fuzzy bags as a generalization of bags. In 2010, the concept of hesitant fuzzy set (HFS), which is another generalization of fuzzy bags, was proposed by Torra [51]. Recently, HFS and its applications are progressing rapidly [13 , 40],

Rosenfeld [41] introduced the concept of fuzzy graphs (FG) based on fundamental idea proposed by Kauffman [23], and obtained some structures related to FGs. Relations between FGs and fuzzy groups were discussed by Bhattacharya [10], and some remarks were given. Bhutani [11] defined concepts of weak isomorphism, co-weak isomorphism and isomorphism between FGs. Then, Mordeson and Peng [33] defined cartesian product, composition, union and join operations of FGs, and studied on some of their properties. Sunitha and Kumar [49] modified definition of complement of a FG, and studied some properties of self complementary FGs. Also some works on FGs may be found in [4 , 47].

The concepts of intuitionistic fuzzy relation (IFR) and intuitionistic fuzzy graph (IFG) were defined by Atanassov [7]. Based on a special case of Atanassov’s definition, Karunambigai and Parvathi [18] introduced IFG, and defined operations between IFGs. Akram and Davvaz [2] investigated some properties of IFGs. Sahoo and Pal [45] defined some product operations on IFGs and discussed some of their properties. Some more works on IFRs, IFGs and mean operators of IFSs may be found in [1 , 48].

In 2015, Pathinathan et al. [39] constructed a new graph structure called hesitancy fuzzy graph (HFG), and gave some basic concepts related to this structure. Even though Pathinathan et al. [39] introduce the concept of hesitancy fuzzy graph, they don’t assign hesitant fuzzy elements (HFEs) to vertices and edges of graph. They use IF-values instead of HFEs, and indicate these IF-values with triples including membership degree, hesitancy degree and non-membership degree of vertices and edges. It can be seen that Pathinathan et al.’s definition is a similar structure to neutrosophic graphs [3 , 17] with some aspects.

Despite the fact that the concept of HFS is a similar concept to the IFSs, there are some basic differences between them in terms of their interpretations and operations. The HFS is more functional than IFS in a decision making problem to model hesitancy in opinion over objects. In a similar way, the HFG is a generalization of the IFG and FG, but it is a more useful tool than IFGs for modeling some decision making problems which include hesitancy of decision makers related to vertices and edges.

Taking into account these considerations, in this paper, we define some new concepts such as Cartesian product of HFSs, hesitant fuzzy relation, hesitant fuzzy graph (HFG), which is different from definition in [39], hesitant fuzzy subgraph (HF-subgraph), partial HF-subgraph by using definition of HFG structure defined in this paper, and give examples related to these concepts. We also introduce concepts of homomorphism, isomorphism, weak isomorphism and co-weak isomorphism between two HFGs based on definition of HFG. Furthermore we define operations of Cartesian product, direct product, lexicographical product and strong product between two HFGs and give examples related these operations, and obtain some of their properties. Finally, to determine dominant element in a group an application of the HFGs in decision making is given.

2. Preliminaries

In this section, we recall basic definitions related to graphs and HFSs.

2.1 2.1. Graphs

A graph is an ordered pair G (V, E), where V is the set of vertices of G and E is the set of edges, formed by pairs of vertices. A simple graph is an undirected graph that has no loop and no more than one edge between any two different vertices. Throughout the paper, G will be considered as a simple graph. Let G = (V, E) be a graph. If for any x, y ∈ V, {x, y} is an edge of graph G it is said that vertices x and y are adjacent in G. Let G = (V, E) and H = (U, F) be two graphs. If U ⊆ V and F ⊆ E, then it is said to be graph H is a subgraph of graph G.

Let G₁ = (V₁, E₁) and G₂ = (V₂, E₂) be two graphs.

The graph G₁ × G₂ = (V, E) is called the cartesian product of G₁ and G₂, where V = (V₁ × V₂) and E = {(x, x₂) (x, y₂) |x ∈ V₁, x₂y₂ ∈ E₂} ∪ {(x₁, z) (y₁, z) |z ∈ V₂, x₁y₁ ∈ E₁} .

The graph G₁ ∗ G₂ = (V, E) is called the direct product of G₁ and G₂, where V = (V₁ × V₂) and E = {(x₁, x₂) (y₁, y₂) |x₁y₁ ∈ E₁, x₂y₂ ∈ E₂} .

The graph G₁ • G₂ = (V, E) is called the lexicographic product of G₁ and G₂, where V = (V₁ × V₂) and E = {(x, x₂) (y, y₂) |x ∈ V₁, x₂y₂ ∈ E₂} ∪ {(x₁, x₂) (y₁, y₂) |x₁y₁ ∈ E₁, x₂y₂ ∈ E₂} .

Let G₁ = (V₁, E₁) and G₂ = (V₂, E₂) be two graphs. The graph G₁ ⊗ G₂ = (V, E) is called the strong product of G₁ and G₂, where V = (V₁ × V₂) and E = {(x, x₂) (y, y₂) | x ∈ V₁, x₂y₂ ∈ E₂} ∪ {(x₁, z) (y₁, z) |z ∈ V₂, x₁y₁ ∈ E₁} ∪ {(x₁, x₂) (y₁, y₂) |x₁y₁ ∈ E₁, x₂y₂ ∈E₂} .

2.2. Hesitant fuzzy sets

Definition 2.1. [51] Let X be a non-empty set. Then, a hesitant fuzzy set (shortly HFS) in X is in terms of a function that when applied to X return a subset of [0, 1] . We express the HFS by a mathematical symbol: A = {(x, ξ_A (x)) : x ∈ X} , where ξ_A (x) is a set of some values in [0, 1] , denoting the possible membership degrees of the element x ∈ X to the set A, ξ = ξ_A (x) is called a hesitant fuzzy element (HFE) and $H (X)$ denotes the set of all HFEs on X.

Definition 2.2. [51] Let $A \in H (X)$ . Then, lower and upper bound of ξ_A (x) related to x ∈ A are defined as follows: $ξ_{A}^{-} (x) = min ξ_{A} (x) and ξ_{A}^{+} (x) = max ξ_{A} (x),$ respectively.

Definition 2.3. [51] Let $A \in H (X)$ . α-lower and α-upper bounds of HFS A are defined as follows: $A_{α}^{-} = {(x, ξ_{A_{α}}^{-} (x)) : x \in A}$ $A_{α}^{+} = {(x, ξ_{A_{α}}^{+} (x)) : x \in A},$ respectively. Here, $ξ_{A_{α}}^{-} (x) = {γ \in ξ_{A} (x) : γ \leq α}$ and $ξ_{A_{α}}^{+} (x) = {γ \in ξ_{A} (x) : γ \geq α} .$

Definition 2.4. [51] Let $A, B \in H (X)$ . Then, complement of HFS, union and intersection operations between two HFSs are defined as follows:

$ξ_{A}^{c} (x) = {1 - γ : γ \in ξ_{A} (x)};$

$(ξ_{A} \cup ξ_{B}) (x) = {γ \in ξ_{A} (x) \cup ξ_{B} (x) : γ \geq max {ξ_{A}^{-} (x), ξ_{B}^{-} (x)}};$

$(ξ_{A} \cap ξ_{B}) (x) = {γ \in ξ_{A} (x) \cup ξ_{B} (x) : γ \leq min {ξ_{A}^{+} (x), ξ_{B}^{+} (x)}} .$

Definition 2.5. [53] Let $A \in H (X)$ , and ξ_A (x) be a HFE of HFS A. Then, $δ (ξ_{A} (x)) = \frac{1}{l (ξ_{A} (x))} \sum_{γ \in ξ_{A} (x)} γ$ is called the score function of ξ_A (x).

Here l (ξ_A (x)) denotes the number of values in ξ_A (x) .

Deepak and John [14] defined hesitant fuzzy subset (HFS-subset) of a HFS as follows:

Definition 2.6. [14] Let $A, B \in H (X)$ . If, for all x ∈ X, δ (ξ_A (x)) ≤ δ (ξ_B (x)), then it is said that HFS A is a HFS-subset of HFS B, and denoted by A ⪯ B.

Example 1. Let X = {x₁, x₂, x₃} be a discourse set, HFSs A and B on X be A = {(x₁, {0.3, 0.4, 0.5}) , (x₂, {0.4, 0.8, 0.6, 0.7}) , (x₃, {0.5, 0.2})} and B = {(x₁, {0.5, 0.9}) , (x₂, {0.8, 1.0, 0.9}) , (x₃, {0.6, 0.7, 0.8})}. It is easy verified that A ⪯ B. Also, by Definition 2.6, δ (ξ_A (x₁)) =0.4 ≤ 0.7 = δ (ξ_B (x₁)), δ (ξ_A (x₂)) ≤ δ (ξ_B (x₂)) , and δ (ξ_A (x₃)) ≤ δ (ξ_B (x₃)). Therefore, we say that A ⪯ B.

Based on score based intersection given in [14], we define score based intersection operation as follows.

Definition 2.7. Let $A, B \in H (X)$ represented by their membership functions ξ_A and ξ_B. Then, a score based intersection of HFEs ξ_A (x) and ξ_B (x), denoted by $ξ_{A} (x) \tilde{\land} ξ_{B} (x)$ is defined as follows: $(ξ_{A} \tilde{\land} ξ_{B}) (x) = {\begin{matrix} ξ_{A} (x), & δ (ξ_{A} (x)) < δ (ξ_{B} (x)) \\ ξ_{B} (x), & δ (ξ_{B} (x)) < δ (ξ_{A} (x)) \\ ξ_{A} (x) or ξ_{B} (x), & δ (ξ_{B} (x)) = δ (ξ_{A} (x)) \end{matrix}$

Definition 2.8. Let $A, B \in H (X)$ . If, for all x ∈ X, $ξ_{A}^{+} (x)) \leq ξ_{B}^{-} (x)$ , then it is said that HFS A is a strong hesitant fuzzy subset (SHFS-subset) of HFS B, and denoted by A ≺ B.

Proposition 2.9. If HFS A is a SHFS-subset of HFS B, then for all x ∈ X, δ (ξ_A (x)) ≤ δ (ξ_A (x)) .

Proof. The proof is easily made by using definition of SHFS-subset and scores of HFS.□

Proposition 2.10. Let A and B be two HFSs over X. Then, $δ (ξ_{A} (x) \tilde{\land} ξ_{B} (x)) = δ (ξ_{A} (x)) \land δ (ξ_{B} (x))$ .

Proof. The proof is obvious from Definition.□

3 3. Hesitant fuzzy relation

Definition 3.1. Let A and B be two HFSs over sets X and Y, respectively. Then, the Cartesian product of HFSs A and B, denoted by $A \tilde{\times} B$ , is defined as follows: $A \tilde{\times} B = {((x, y), ξ_{A} (x) \tilde{\land} ξ_{B} (y)) : (x, y) \in X \times Y}$

Definition 3.2. Let X and Y be two non-empty sets and let A and B be two HFSs over sets X and Y, respectively. Then, hesitant fuzzy relation (shortly HFR) $R$ from the HFS A into HFS B, denoted by $ξ_{R}$ , is a HFS on X × Y such that $δ (ξ_{R} (x, y)) \leq δ (ξ_{A} (x)) \land δ (ξ_{B} (y))$ , for all (x, y) ∈ X × Y.

If A = B, then $R$ is called a HFR on HFS A.

Example 2. Let X = {x₁, x₂, x₃} and Y = {y₁, y₂} and let A and B be two HFSs on X and Y given as follows: Ax₁x₂x₃

δ (ξ_A (x₁)) =0.367, δ (ξ_A (x₂)) =0.550, δ (ξ_A (x₃)) =0.433 and $B = {(y_{1}, {0.4, 0.5, 0.7}), (y_{2}, {0.2, 0.5, 0.6})},$ δ (ξ_B (y₁)) =0.533, δ (ξ_B (y₂)) =0.433, respectively. Then,

$\begin{matrix} A \tilde{\times} B = \\ {((x_{1}, y_{1}), {0.2, 0.4, 0.5}), ((x_{1}, y_{2}), {0.2, 0.4, 0.5}), \\ ((x_{2}, y_{1}), {0.4, 0.5, 0.7}), ((x_{2}, y_{2}), {0.2, 0.5, 0.6}), \\ ((x_{3}, y_{1}), {0.1, 0.5, 0.7}), ((x_{3}, y_{2}), {0.2, 0.5, 0.6, 0.7})} \end{matrix}$ and a HFR $R$ can be written as follows: $\begin{matrix} R = \\ {((x_{1}, y_{1}), {0.1, 0.2, 0.4, 0.5}), ((x_{1}, y_{2}), {0.1, 0.2, 0.7}), \\ ((x_{2}, y_{1}), {0.1, 0.2, 0.5, 0.7}), ((x_{2}, y_{2}), {0.3, 0.4}), \\ ((x_{3}, y_{1}), {0.2, 0.3, 0.4, 0.8}), ((x_{3}, y_{2}), {0.1, 0.3, 0.8})} . \end{matrix}$

4. Hesitant Fuzzy Graphs

Definition 4.1. Let G = (V, E) be a graph, $A$ be a HFS on V and $G$ be a HFS on E ⊆ V × V. If $δ (ξ_{G} (x, y)) \leq δ (ξ_{A} (x)) \land δ (ξ_{A} (y))$ for all (x, y) ∈ E, it is called a hesitant fuzzy graph (HFG) over graph G = (V, E) and denoted by $\tilde{G} = (A, G)$ .

HFS $A$ over V is HF-vertex set of HFG and HFS $G$ over E is the HF-edge set of HFG. Note that $G$ is a symmetric HFR on $A$ .

From now on we use the notation xy for (x, y) ∈ E. In this study, we will not appear elements ξ_A (x) = {0} in HFG. Throughout this paper, set of all HFGs over vertex set V will be denoted by $HFG (V)$ .

Example 3. Let V = {v₁, v₂, v₃, v₄, v₅} be an initial universe,

A

be a HFS over V and

G

be a HFS over E = {v₁v₂, v₁v₃, v₁v₅, v₂v₃, v₃v₄, v₃v₅, v₄v₅} ⊆ V × V given as follows:

	v ₁	v ₂	v ₃
ξ _A	{0.2, 0.4, 0.8}	{0.5, 0.6}	{0.1, 0.4, 0.7, 0.9}
Scores	0.467	0.550	0.525

v ₄	v ₅
{0.3, 0.4, 0.5}	{0.2,0.5}
0.400	0.350

	v ₁ v ₂	v ₁ v ₃	v ₁ v ₅
$ξ_{M}$	{0.3, 0.5}	{0.2, 0.4, 0.5}	{0.1, 0.2, 0.3, 0.4}
Scores	0.400	0.367	0.250

v ₂ v ₃	v ₃ v ₄	v ₃ v ₅	v ₄ v ₅
{0.2, 0.3, 0.4}	{0.15, 0.4}	{0.1, 0.35, 0.45}	{0.25, 0.35}
0.300	0.275	0.300	0.300

Then, HFG $\tilde{G} = (A, G)$ can be depicted as in Fig. 1.

Fig.1

$\tilde{G} = (A, G)$ HFG.

Fig.2

HF-subgraph of HF-graph $\tilde{G}$ .

Fig.3

Partial HF-subgraph of HF-graph $\tilde{G}$ .

Definition 4.2. Let $\tilde{G} = (A, M)$ be a HFG on graph $\tilde{G} = (V, E)$ . The degree of a vertex x ∈ V in a HFG $G = (A, M)$ is defined as $d_{\tilde{G}} (x) = \sum_{x, y \neq x \in V} δ (ξ_{M} (xy)) .$

Example 4. Let us consider HFG $\tilde{G}$ in Example 1. Then, $d_{\tilde{G}} (v_{1}) = 1.02, d_{\tilde{G}} (v_{2}) = 0.7, d_{\tilde{G}} (v_{3}) = 1.25, d_{\tilde{G}} (v_{1}) = 0.58$ and $d_{\tilde{G}} (v_{5}) = 0.85 .$

Definition 4.3.Let ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ be two HFGs over G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. If ξ_A (x) ⪯ ξ_B (x) and $ξ_{M} (xy) ⪯ ξ_{N} (xy)$ , then HFG ${\tilde{G}}_{1} = (A, M)$ is called a hesitant fuzzy subgraph (HF-subgraph) of ${\tilde{G}}_{2} = (B, N)$ and denoted by ${\tilde{G}}_{1} ⪡ {\tilde{G}}_{2} .$

Definition 4.4.Let ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ be two HFGs over G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. If V₁ ⊆ V₂, ξ_A (x) ⪯ ξ_B (x) for all x ∈ V₁ and $ξ_{M} (xy) ⪯ ξ_{N} (xy)$ for all x, y ∈ V₁, then HFG ${\tilde{G}}_{1} = (A, M)$ is called a partial HF-subgraph of ${\tilde{G}}_{2} = (B, N)$ and denoted by ${\tilde{G}}_{1} < {\tilde{G}}_{2} .$

Example 5. Consider HFG $\tilde{G} = (A, G)$ given as in Example. Given HFG ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ as follows:

	v ₁	v ₂	v ₃
ξ _A	{0.2, 0.5}	{0.2, 0.3, 0.6}	{0.1, 0.4, 0.5, 0.8}
Scores	0.350	0.367	0.450

v ₄	v ₅
{0.3, 0.4, 0.5}	{0.1,0.5}
0.400	0.300

	v ₁ v ₂	v ₁ v ₃	v ₁ v ₅
$ξ_{M}$	{0.22, 0.35}	{0.1, 0.3, 0.4}	{0.1, 0.2, 0.3}
Scores	0.285	0.267	0.200

v ₂ v ₃	v ₃ v ₄	v ₃ v ₅	v ₄ v ₅
{0.15, 0.22, 0.33, 0.4}	{0.2, 0.28}	{0.2, 0.25, 0.35}	{0.25, 0.35}
0.367	0.240	0.267	0.300

Then, HFG $\tilde{G} = (A, G)$ can be depicted as in Fig. 1.

	v ₁	v ₂	v ₃	v ₅
ξ _B	{0.2, 0.5}	{0.2, 0.3, 0.5}	{0.1, 0.4, 0.5, 0.6}	{0.1, 0.5}
Scores	0.350	0.333	0.400	0.300

	v ₁ v ₂	v ₁ v ₃	v ₁ v ₅
$ξ_{N}$	{0.2, 0.3}	{0.1, 0.2, 0.3}	{0.1, 0.2}
Scores	0.250	0.200	0.150

v ₂ v ₃	v ₃ v ₅
{0.1, 0.2, 0.3, 0.35}	{0.12, 0.25, 0.32}
0.238	0.230

Then, partial HF-subgraph $\tilde{G} = (B, N)$ can be depicted as in Fig. 2.

Definition 4.5. Let ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ be two HFGs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, bijective mapping φ : V₁ → V₂ satisfying the following conditions is called a homomorphism from ${\tilde{G}}_{1}$ to ${\tilde{G}}_{2}$

δ (ξ_A (x)) ≤ δ (ξ_B (φ (x))),

$δ (ξ_{M} (xy)) \leq δ (ξ_{N} (φ (x) φ (y)))$

for all x ∈ V₁, xy ∈ E₁ ⊆ V₁ × V₁ .

Definition 4.6. Let ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ be two HFGs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, bijective mapping φ : V₁ → V₂ satisfying the following conditions is called an isomorphism from ${\tilde{G}}_{1}$ to ${\tilde{G}}_{2}$

δ (ξ_A (x)) = δ (ξ_B (φ (x))),

$δ (ξ_{M} (xy)) = δ (ξ_{N} (φ (x) φ (y)))$

for all x ∈ V₁, xy ∈ E₁ ⊆ V₁ × V₁ .

Definition 4.7. Let $G_{1} = (A, M)$ and $G_{2} = (B, N)$ be two HFGs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, bijective mapping φ : V₁ → V₂ satisfying the following conditions is called a weak isomorphism from ${\tilde{G}}_{1}$ to ${\tilde{G}}_{2}$ .

f is homomorphism

δ (ξ_A (x)) = δ (ξ_B (φ (x))) ,

for all x ∈ V₁, xy ∈ E₁ ⊆ V₁ × V₁ .

Example 6. Let V₁ = {v₁, v₂, v₃} V₂ = {u₁, u₂, u₃} be two vertex sets and bijective function φ : V₁ → V₂ is defined as φ (v₁) = u₁, φ (v₂) = u₂ and φ (v₃) = u₃. Let ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ be two HFGs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively where ξ_A (v₁) = {0.2, 0.3, 0.5}, ξ_A (v₂) = {0.4, 0.6}, ξ_A (v₃) = {0.3, 0.7, 0.8} and $ξ_{M} (v_{1} v_{2}) = {0.2, 0.4}$ , $ξ_{M} (v_{1} v_{3}) = {0.1, 0.3, 0.5}$ , ξ_B (u₁) = {0.1, 0.2, 0.7}, ξ_B (u₂) = {0.3, 0.5, 0.7}, ξ_B (u₃) = {0.3, 0.6, 0.7, 0.8} and $ξ_{N} (u_{1} u_{2}) = {0.1, 0.2, 0.3},$ $ξ_{N} (u_{1} u_{3}) = {0.15, 0.35}$ .

Here since δ (ξ_A (v₁)) = δ (ξ_B (u₁)), δ (ξ_A (v₂)) = δ (ξ_B (u₂)), δ (ξ_A (v₃)) = δ (ξ_B (u₃)) and $δ (ξ_{M} (v_{1} v_{2})) = δ (ξ_{N} (u_{1} u_{2}))$ , $δ (ξ_{M} (v_{1} v_{3})) = δ (ξ_{N} (u_{1} u_{3}))$ , h is a isomorphism of ${\tilde{G}}_{1}$ onto ${\tilde{G}}_{2}$ .

Remark 1. A weak isomorphism may not be an isomorphism.

In above example, if we get $ξ_{M} (v_{1} v_{2}) = {0.15, 0.30}$ , $ξ_{M} (v_{1} v_{3}) = {0.16, 0.3, 0.5}$ , since $δ (ξ_{M} (v_{1} v_{2})) \neq δ (ξ_{N} (u_{1} u_{2}))$ and $δ (ξ_{M} (v_{1} v_{3}))$ $\neq δ (ξ_{N} (u_{1} u_{3}))$ , then φ is weak isomorphism but φ is not an isomorphism.

Definition 4.8. Let ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ be two HFGs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, bijective mapping φ : V₁ → V₂ satisfying following conditions is called as a co-weak isomorphism from ${\tilde{G}}_{1}$ to ${\tilde{G}}_{2}$ .

φ is homomorphism

$δ (ξ_{M} (xy)) = δ (ξ_{N} (φ (x) φ (y)))$

for all xy ∈ E₁ ⊆ V₁ × V₁ .

Example 7. Let us consider function φ in Example 6. Let ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ be two HFGs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively where ξ_A (v₁) = {0.2, 0.5}, ξ_A (v₂) = {0.4, 0.5, 0.7}, ξ_A (v₃) = {0.3, 0.7} and $ξ_{M} (v_{1} v_{2}) = {0.1, 0.4}$ , $ξ_{M} (v_{1} v_{3}) = {0.2, 0.3, 0.4}$ ,

ξ_B (u₁) = {0.5, 0.7, 0.8}, ξ_B (u₂) = {0.4, 0.8}, ξ_B (u₃) = {0.3, 0.5, 0.9} and $ξ_{N} (u_{1} u_{2}) = {0.2, 0.3}$ $ξ_{N} (u_{1} u_{3} = {0.2, 0.7}$ . Here, since φ is a homomorphism from $(A, M)$ to $(B, N)$ and $δ (ξ_{M} (v_{1} v_{2})) = δ (ξ_{N} (u_{1} u_{2}))$ , $δ (ξ_{M} (v_{1} v_{3})) = δ (ξ_{N} (u_{1} u_{3}))$ , f is a co-weak isomorphism from ${\tilde{G}}_{1}$ to ${\tilde{G}}_{2}$ .

Note that f is not an isomorphism from ${\tilde{G}}_{1}$ to ${\tilde{G}}_{2}$ , since ξ_A (v₁) ≠ξ_B (u₁), δ (ξ_A (v₂)) ≠δ (ξ_B (u₂)) and δ (ξ_A (v₃)) ≠δ (ξ_B (u₃)) .

Definition 4.9. Let ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ be two HFGs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, Cartesian product of two HFGs ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ is denoted by ${\tilde{G}}_{1} \hat{\times} {\tilde{G}}_{2} = (A \times B, M \times N)$ and defined as follows:

$(ξ_{A} \times ξ_{B}) (x, y) = ξ_{A} (x) \tilde{\land} ξ_{B} (y)$ for all (x, y) ∈ V₁ × V₂,

$(ξ_{M} \times ξ_{N}) ((x, y) (x, z)) = ξ_{A} (x) \tilde{\land} ξ_{N} (yz)$ for all x ∈ V₁, yz ∈ E₂.

$(ξ_{M} \times ξ_{N}) ((x, z) (y, z)) = ξ_{M} (xy) \tilde{\land} ξ_{B} (z)$ for all z ∈ V₂, xy ∈ E₁.

Example 8.Consider two HFGs ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ given as in Fig. 4. Then, Cartesian product of HFGs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is as in Fig. 5.

Fig.4

${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ HFGs, respectively.

Fig.5

HFG ${\tilde{G}}_{1} \hat{\times} {\tilde{G}}_{2}$ .

Proposition 4.10. Let ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ be two HFGs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, ${\tilde{G}}_{1} \hat{\times} {\tilde{G}}_{2}$ is an HFG.

Proof. In Definition 4.9 Case 1 is clear form definition of Cartesian product of two HFSs. Let x ∈ V₁, yz ∈ E₂. Since $\begin{matrix} (ξ_{M} \times ξ_{N}) ((x, y) (x, z)) = ξ_{A} (x) \land ξ_{N} (yz), \\ δ (ξ_{M} \times ξ_{N}) ((x, y) (x, z))) = δ (ξ_{A} (x) \tilde{\land} ξ_{N} (yz)) \\ \leq δ (ξ_{A} (x)) \land δ ((ξ_{B} (y) \land ξ_{B} (z))) \\ = (δ (ξ_{A} (x)) \land δ (ξ_{B} (y))) \land (δ (ξ_{A} (x)) \land δ (ξ_{B} (z))) \\ = δ ((ξ_{A} \times ξ_{B}) (x, y)) \land δ ((ξ_{A} \times ξ_{B}) (x, z)) . \end{matrix}$ Let z ∈ V₂, xy ∈ E₁. Then, $\begin{matrix} (ξ_{M} \times ξ_{N}) ((x, z) (y, z)) = ξ_{M} (xy) \tilde{\land} ξ_{B} (z), \\ δ (ξ_{M} \times ξ_{N}) ((x, z) (y, z))) = δ (ξ_{M} (xy) \tilde{\land} ξ_{B} (z)) \\ \leq δ ((ξ_{A} (x) \tilde{\land} ξ_{A} (y))) \land δ (ξ_{B} (z)) \\ = (δ (ξ_{A} (x)) \land δ (ξ_{B} (z))) \land (δ (ξ_{A} (y)) \land δ (ξ_{B} (z))) \\ = δ ((ξ_{A} \times ξ_{B}) (x, z)) \land δ ((ξ_{A} \times ξ_{B}) (y, z)) . \end{matrix}$ Therefore, ${\tilde{G}}_{1} \hat{\times} {\tilde{G}}_{2}$ is an HFG.□

Definition 4.11. Let ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ be two HFGs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, direct product of two HFGs ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ is denoted by ${\tilde{G}}_{1} * {\tilde{G}}_{2} = (A * B, M * N)$ and defined as follows:

$(ξ_{A} * ξ_{B}) (x, y) = ξ_{A} (x) \tilde{\land} ξ_{B} (y)$ , for all (x₁, x₂) ∈ V₁ × V₂,

$(ξ_{M} * ξ_{N}) ((x, y) (z, t)) = ξ_{M} (xz) \tilde{\land} ξ_{N} (yt),$ for all xz ∈ E₁ and yt ∈ E₂.

Example 9. Consider two HFGs ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ given in Example 8. Then, direct product of HFGs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is as in Fig. 6.

Fig.6

HFG $G_{1} * G_{2}$ .

Proposition 4.12. Let ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ be two HFGs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, ${\tilde{G}}_{1} * {\tilde{G}}_{2}$ is a HFG.

Proof. The proof can be made in similar way to proof of Proposition 4.10.

Definition 4.13. Let ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ be two HFGs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, lexicographical product of two HFGs ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ is denoted by ${\tilde{G}}_{1} • {\tilde{G}}_{2} = (A • B, M • N)$ and defined as follows:

(ξ_A • ξ_B) (x, y) = (ξ_A (x) ∧ ξ_B (y)) , for all (x, y) ∈ V₁ × V₂),

$(ξ_{M} • ξ_{N}) ((x, y) (x, z)) = ξ_{A} (x) \land ξ_{N} (yz)$ , for all yz ∈ E₂),

$(ξ_{M} • ξ_{N}) ((x, y) (z, t)) = (ξ_{M} (xz), ξ_{N} (yt))$ , for all xz ∈ E₁ and yt ∈ E₂.

Example 10. Consider two HFGs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ given in Example 8. Then, lexicographic product of HFGs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is as in Fig. 7.

Fig.7

HFG ${\tilde{G}}_{1} • {\tilde{G}}_{2}$ .

Proposition 4.14. Let ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ be two HFGs. Then, ${\tilde{G}}_{1} • {\tilde{G}}_{2}$ is a HFG

Proof. The proof is can be made by similar way to proofs of Proposition 4.10.□

Definition 4.15. Let ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ be two HFGs over graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂), respectively. Then, strong product of two HFGs ${\tilde{G}}_{1} = (A, M)$ and ${\tilde{G}}_{2} = (B, N)$ is denoted by ${\tilde{G}}_{1} \otimes {\tilde{G}}_{2} = (A \otimes B, M \otimes N)$ and defined as follows:

$(ξ_{A} \otimes ξ_{B}) (x, y) = ξ_{A} (x) \tilde{\land} ξ_{B} (y),$ for all (x, y) ∈ V₁ × V₂,

$(ξ_{M} \otimes ξ_{N}) ((x, y) (x, z)) = ξ_{A} (x) \tilde{\land} ξ_{N} (yz)$ , for all yz ∈ E₂,

$(ξ_{M} \otimes ξ_{N}) ((x, z) (y, z)) = ξ_{N} (xy) \tilde{\land} ξ_{B} (z)$ , for all xy ∈ E₁ and z ∈ V₂.

$(ξ_{M} \otimes ξ_{N}) ((x, y) (z, t)) = ξ_{M} (xz) \tilde{\land} ξ_{N} (yt)$ , for all xz ∈ E₁ and yt ∈ E₂.

Example 11. Consider two HFGs ${\tilde{G}}_{1} = (μ, η)$ and ${\tilde{G}}_{2} = (μ^{'}, η^{'})$ given in Example 8. Then, strong product of HFGs ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is as in Fig. 8.

Fig.8

HFG ${\tilde{G}}_{1} \otimes {\tilde{G}}_{2}$ .

Proposition 4.16. Let ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ be two HFGs. Then, ${\tilde{G}}_{1} \otimes {\tilde{G}}_{2}$ is a HFG.

Proof. It is similar to Proposition 4.10 and 4.12.□

Corollary 4.17. Let ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ be two HFGs. Then,

${\tilde{G}}_{1} • {\tilde{G}}_{2} ⪡ {\tilde{G}}_{1} \otimes {\tilde{G}}_{2}$ ,

${\tilde{G}}_{1} * {\tilde{G}}_{2} ⪡ {\tilde{G}}_{1} \otimes {\tilde{G}}_{2}$ ,

${\tilde{G}}_{1} \times {\tilde{G}}_{2} ⪡ {\tilde{G}}_{1} \otimes {\tilde{G}}_{2}$ ,

${\tilde{G}}_{1} * {\tilde{G}}_{2} ⪡ {\tilde{G}}_{1} • {\tilde{G}}_{2} .$

Proof. The proofs are clear from Definitions 4.3 and 4.4.□

5. Application of HFGs in Decision Making

In this section, some definitions to be used in suggested method are presented. Decision making method is developed and given its algorithm in order to choose element which has dominate degree in group, and to show process of proposed decision making method we give an illustrative example.

Definition 5.1. [52] Let ξ_i (i = 1, 2,. . . , n) be a collection of HFEs. Then, hesitant fuzzy weighted averaging operators (HFWA) is a mapping HFWA : HFEⁿ → HFE such that $H F W A_{w} (ξ_{1}, ξ_{2}, ..., ξ_{n}) = \oplus_{i = 1}^{n} w_{i} ξ_{i}$ (1) $= \underset{α_{1} \in ξ_{1}, α_{1} \in ξ_{2}, ..., α_{n} \in ξ_{1}}{\cup} {1 - \prod_{i = 1}^{n} {(1 - α_{i})}^{w_{i}}}$ where w = (w₁, w₂,. . . , w_n) ^T is the weight vector of ξ_i (i = 1, 2,. . . , n), with w_i ≥ 0 (i = 1, 2,. . . , n) and $\sum_{j = 1}^{n} w_{j} = 1$ (see [52]). Here if it is taken all of w_i (i = 1, 2,. . . , n) as equal, then Eq. (1) can be written as follows:

$H F W A (ξ_{1}, ξ_{2}, ...,_{n}) = \frac{1}{n} \oplus_{i = 1}^{n} ξ_{i} .$ (2)

Eq. (2) can be written as more explicit, respectively, as follows:

$\oplus_{i = 1}^{n} ξ_{i} = ⋃_{α_{1} \in ξ_{1}, α_{1} \in ξ_{2}, . . ., α_{n} \in ξ_{1}} {1 - \prod_{i = 1}^{n} (1 - α_{i})^{\frac{1}{n}}}$

Based on Definition 5.1 and 2.5, we define following concepts:

Definition 5.2. Let ${\tilde{G}}_{1} = (A, M)$ be a directed HFGs over graphs G = (V, E) and v_r, r = 1, 2,. . . , n be adjacent HF-vertices of v_k ∈ V. Out-degree of v_k, denoted by $O_{d} (ν_{k})$ , is defined as follows:

$O_{d} (v_{k}) = ⋃_{v_{k} v_{r} \in ξ_{M}} {1 - \prod_{r = 1}^{n} (1 - ξ_{M} (v_{k} v_{r}))},$ (3) $ℐ$ Similarly, in-degree of v_k, denoted by $ℐ_{d} (ν_{k})$ is defined as follows:

$ℐ_{d} (ν_{k}) \underset{v_{r} v_{k} \in ξ_{ℳ}}{\cup} {1 - \prod_{r = 1}^{n} (1 - ξ_{ℳ} (v_{r} v_{k}))}$ (4)

Scores of out-degree and in-degree of a vertex v_k, dednoted by $δ (O_{d} (v_{k}))$ and $δ (I_{d} (v_{k}))$ respectively, is calculated by using Definitions 2.5.

Let $δ (O_{d} (v_{k}))$ and $δ (I_{d} (v_{k}))$ be scores of out-degree and in-degree of a vertex v_k, respectively. Then $δ (O_{d} (v_{k})) - δ (I_{d} (v_{k}))$ is called domination degree of vertex v_k and denoted by $D_{ditsc} (v_{kitsc})$ .

Algorithm 1

Input: The directed hesitant fuzzy graph
Output: Dominate element
algorithmic
1. Input the set of vertices V = {v₁, v₂,. . . , v_n} and a HFS A which is defined on set V
2. Input the set of edges E = {e₁, e₂,. . . , e_n} .
3. Compute the membership degrees of edge using relation $δ (ξ_{M} (xy)) \leq δ (ξ_{A} (x)) \land δ (ξ_{A} (y))$ .
4. Compute out-degree and in-degree for each of vertices by using Eqs. (3) and (4).
5. Find score values of out-degree and in-degree for each of vertices.
6. Find domination degree for each of vertices.
7. Rank vertices according to real domination degrees of them.
8. Choose vertex which has maximum domination degrees.

5.1. Illustrative Example

Let us consider the application in [3, 17]. There are seven persons which are influence each other on a social group on ICQ. Let P = {p₁, p₂, p₃, p₄, p₅, p₆, p₇} be set of seven person in a social group on ICQ.

Step 1: $A = {(p_{1}, {0.4, 0.5, 0.9}), (p_{2}, {0.3, 0.6}),$ (p₃, {0.1, 0.5, 0.6, 0.7}) , (p₄, {0.35, 0.40}) , (p₅, {0.45, 0.8, 0.75}) , (p₆, {0.6, 0.9}) , (p₇, {0.2, 0.55, 0.8}) is the HFS on the set V, where these HFEs related to persons represent his good influence on others.

	p ₁	p ₂	p ₃	p ₄	p ₅	p ₆	p ₇
Scores	0.600	0.450	0.475	0.375	0.667	0.750	0.517

Step 2: J = {(p₆, p₃) , (p₆, p₄) , (p₆, p₇) , (p₆, p₂) , (p₁, p₃) , (p₂, p₃) , (p₇, p₁) , (p₅, p₄) , (p₅, p₂) , (p₂, p₇)} is the set of edges.

Step 3: Let

M

be HFS on the set E as shown in following table.

Edge	hesitant fuzzy values	scores
(p₁, p₃)	{0.3, 0.4, 0.5}	0.400
(p₁, p₆)	{0.5, 0.6}	0.550
(p₁, p₇)	{0.2, 0.3, 0.7}	0.400
(p₂, p₁)	{0.1, 0.5, 0.7}	0.433
(p₂, p₃)	{0.2, 0.3, 0.4, 0.5}	0.350
(p₃, p₇)	{0.45}	0.450
(p₃, p₄)	{0.1, 0.5}	0.300
(p₄, p₇)	{0.1, 0.2, 0.3}	0.200
(p₅, p₄)	{0.1, 0.4, 0.6}	0.367
(p₅, p₂)	{0.4, 0.5}	0.450
(p₆, p₃)	{0.2, 0.3, 0.5, 0.7}	0.425
(p₆, p₄)	{0.1, 0.2}	0.15
(p₆, p₇)	{0.2, 0.6, 0.7}	0.500
(p₆, p₂)	{0.2, 0.5}	0.350
(p₇, p₅)	{0.3, 0.4, 0.5, 0.8}	0.500

HFS B of edges

The HF-diagraph $\tilde{G} = (A, M)$ is shown in Fig. 9.

Fig.9

HF-diagraph.

Step 4: By using Eqs. (3) and (4), out-degrees and in-degrees of each person are obtained as follows:

$\begin{matrix} O_{d} (p_{1}) & = & {0.79, 0.8, 0.832}, \\ O_{d} (p_{2}) & = & {0.865, 0.832}, \\ O_{d} (p_{3}) & = & {0.45, 0.55}, \\ O_{d} (p_{4}) & = & {0.496}, \\ O_{d} (p_{5}) & = & {0.784, 0.7}, \\ O_{d} (p_{6}) & = & {0.916, 0.28, 0.904, 0.6}, \\ O_{d} (p_{7}) & = & {0.958}, \end{matrix}$

$\begin{matrix} I_{d} (p_{1}) & = & {0.865} \\ I_{d} (p_{2}) & = & {0.7, 0.6} \\ I_{d} (p_{3}) & = & {0.79, 0.832, 0.916} \\ I_{d} (p_{4}) & = & {0.55, 0.784, 0.28} \\ I_{d} (p_{5}) & = & {0.958} \\ I_{d} (p_{6}) & = & {0.8} \\ I_{d} (p_{7}) & = & {0.832, 0.45, 0.496, 0.904} . \end{matrix}$

Step 5: By using Eq. (2.5), scores of out-degrees and in-degrees of each person are obtained as follows:

$δ (O_{d} (p_{1})) = 0.807, δ (I_{d} (p_{1})) = 0.865$

$δ (O_{d} (p_{2})) = 0.849, δ (I_{d} (p_{2})) = 0.650$

$δ (O_{d} (p_{3})) = 0.500, δ (I_{d} (p_{3})) = 0.846$

$δ (O_{d} (p_{4})) = 0.496, δ (I_{d} (p_{4})) = 0.538$

$δ (O_{d} (p_{5})) = 0.742, δ (I_{d} (p_{5})) = 0.958$

$δ (O_{d} (p_{6})) = 0.675, δ (I_{d} (p_{6})) = 0.800$

$δ (O_{d} (p_{7})) = 0.958, δ (I_{d} (p_{7})) = 0.671 .$

Step 6: Real domination degrees of each person are obtained as follows: $\begin{matrix} D_{d} (p_{1}) & = & 0.807 - 0.865 = - 0.058 \\ D_{d} (p_{2}) & = & 0.849 - 0.645 = 0.196 \\ D_{d} (p_{3}) & = & 0.500 - 0.846 = - 0.346 \\ D_{d} (p_{4}) & = & 0.496 - 0.538 = - 0.042 \\ D_{d} (p_{5}) & = & 0.742 - 0.958 = - 0.216 \\ D_{d} (p_{6}) & = & 0.675 - 0.800 = - 0.125 \\ D_{d} (p_{7}) & = & 0.958 - 0.6705 = 0.2875 . \end{matrix}$

Step 7: Domination degrees of persons in this group can be rank in form p₇ ≥ p₂ ≥ p₄ ≥ p₁ ≥ p₆ ≥ p₅ ≥ p₃ .

Step 8: p₇ is a person who has real dominant effect in the group.

6. Conclusion

In this paper, definition of HFG, some concepts and operations related to HFGs such as HF-subgraph, partial HF-subgraph, homomorphism, isomorphism, weak-isomorphism and co-weak isomorphism between HFGs are first given. Also, some product operations are defined, and some properties of them are obtained. Furthermore, to find dominant element in a directed HFG, a method is proposed and is given its an application in decision making. In the future, other operations and definitions existing in fuzzy graphs and classical graphs may be extended to HFGs, and also researchers can study on applications of HFGs in decision making and clustering analysis based on defined new concepts and operations.

Conflict of interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Footnotes

Acknowledgement

The author is very grateful to anonymous reviewers for their valuable comments and constructive suggestions, which greatly improved the quality of this paper.

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