Hyper-Wiener index for fuzzy graph and its application in share market

Abstract

Topological indices have an important role in molecular chemistry, network theory, spectral graph theory and several physical worlds. Most of the topological indices are defined in a crisp graph. As fuzzy graphs are more generalization of crisp graphs, those indices have more application in fuzzy graphs also. In this article, we introduced the fuzzy hyper-Wiener index (FHWI) and studied this index for various fuzzy graphs like path, cycle, star, etc and provided some interesting bounds of FHWI for that fuzzy graph. A lower bound of FHWI is established for n-vertex connected fuzzy graph depending on strength of a strong edges. A relation between FHWI of a tree and its maximum spanning tree is established and this index is calculated for the saturated cycle. Also, at the end of the article, an application in the share market of this index is presented.

Keywords

Fuzzy graph wiener index hyper-wiener index fuzzy hyper-wiener index

Abbreviations

FSS

Fuzzy subset

Fuzzy graph

FSG

Fuzzy subgraph

PFSG

Partial fuzzy subgraph

CPFG

Complete fuzzy graph

Topological index

Wiener index

GDS

Geodesic

HWI

Hyper-Wiener index

FHWI

Fuzzy hyper-Wiener index

Membership value

1 Introduction

1.1 Research background

Rosenfeld [30] first introduced the concept of fuzzy graph (FG) and also provided the fuzzy relation (FR), fuzzy bridge, fuzzy block and fuzzy distances of a FG. In that time, Yeh et al. [36] also established FG separately and gave an application of FG in clustering analysis. In [4 , 24–29], Rashmanlou et al. studied antipodal intervalued FG, balanced interval valued FG, product vague graph, domination in vague graphs, intuitionistic FG, categorial properties and product of bipolar FG, etc. In [20] Poulik and Ghorai studied empirical Results on Operations of Bipolar FG and in [22] Poulik, Ghorai and Xin studied Pragmatic results on interval-valued FG and interval-valued neutrosophic graph. In 1, 13–15, 17, 18, 31–34, one can see for more generalization on FGs.

Topological indices (TIs) are molecular descriptors and applied in chemistry, spectral graph theory, network theory. Such indices are representatives of chemical compounds which describe the structure of that molecular graph. In the Molecular graph, vertex represents atom and edge represents the bond between two atoms. In 1947 [35], Harold Wiener first introduced the Wiener index (WI) which has a huge application in chemical graph theory and spectral graph theory. In the same paper, he proved that the boiling point of paraffins depends on WI. In [5], one can see for further work-related this index. In [23], Randic introduced hyper-Wiener index (HWI) which is a generalization of WI and used in biochemistry, chemical graph theory, spectral graph theory. HWI has studied for both theoretical and application points of view also. In [11, 16], one can see other TIs. Binu et al. [2, 3], introduced connectivity index an Wiener index of a FG. Also, In 2021, Fang et al. [6], introduced connectivity index and Wiener index of a fuzzy incidence graph. In [7] Islam, Maity and Pal studied the Wiener index for the saturated fuzzy cycle. Kalathian et al. [10] studied some topological indices for fuzzy graphs in 2020. Islam and Pal introduced First Zagreb index and F-index for fuzzy graph in [8, 9] respectively. In [12], Mahapatra, Samanta and Pal provided the RSM index in FG. Poulik and Ghorai studied the Certain index and Wiener absolute index for a bipolar fuzzy graph in [19, 21].

1.2 Objective of the work

Different types of topological index of a graph has many applications and many results are available for crisp graphs.

But in many practical applications it is seen that many situations cannot be modelled using crisp graphs.

In these cases, to handle such a situation, those topological indices are needed to define a fuzzy graph. In this article, a fuzzy hyper-Wiener index (FHWI) is introduced which is a more generalization of WI, HWI, FWI. As FHWI is defined on a fuzzy graph, this index provided the better result in application. The value of this index for different types of FG like path, cycle, star, etc. are studied. In the end of the article, a decision making problem in the share market is presented and solved by using the FHWI.

1.3 Framework of the article

Structure of the article is as follows: In sec. 2, some basic definitions and useful results are provided which are essential to the development of our content. In sec. 3, FHWI is defined and studied for several FG like as path, cycle, star, etc. In sec. 4, a decision making problem in the share market is presented and solved by using the FHWI.

2 Preliminaries

Some basic definitions and useful results are provided in this section, most of them one can be found in [18].

Let X (≠ φ) be a given finite set. Now the FG is G = (θ, ρ), where θ is fuzzy subset(FSS) of X and ρ is FSS of X × X with ρ (x, y) ≤ ∧ {θ (x) , θ (y)}, where ∧ indicates the minimum. It is noted that ρ is called a FR on θ. It is assumed that ρ is reflexive as well as symmetric. We write G^* = (θ^*, ρ^*), where θ^* = {x ∈ X : θ (x) ≠0} and ρ^* = {(x, y) ∈ X × X : ρ (x, y) ≠0}. Here θ^* and ρ^* is known as the vertex set and edge set of the FG, respectively. The fuzzy graph G is trivial if G^* is trivial. Now H = (φ, ω) is called partial fuzzy subgraph(PFSG) of G if for all x ∈ φ^*, φ (x) ≤ θ (x) and for all (x, y) ∈ ω^*, ω (x, y) ≤ ρ (x, y). If for all x ∈ φ^*, φ (x) = θ (x) and for all (x, y) ∈ ω^*, ω (x, y) = ρ (x, y), then H is called FSG of G. A FSG, H = (φ, ω) spans the FG G = (θ, ρ) if φ = θ. It is noted that G_xy is a FSG of G with ω (x, y) =0 and G_x is FSG of G where φ (x) =0. Let x₀, x₁, ⋯ , x_n be distinct vertices in G. Then we call P = x₀x₁ ⋯ x_n is path in G if ρ (x_i, x_i+1) ≠0 for i = 0, 1, ⋯ , n - 1. For this case we call the length of the path n. The path, P is called a cycle if ρ (x₀, x_n) >0 . Let x₀, x₁, ⋯ , x_n be the vertices of G. Then G = (x₀, x₁, x₁ ⋯ x_n) is called a star if ρ (x₀, x_i) ≠0 for i = 0, 1, ⋯ , n. Here x₀ is called the center of the star. G is called a complete FG (CPFG) if for all x, y ∈ θ^*, ρ (x, y) = θ (x) ∧ θ (y) , where ∧ denotes the minimum. Let P = x₀x₁ ⋯ x_n be a path in G. Then strength of the path P, S (P) is defined by S (P) = ∧ {ρ (x_i, x_i+1) : i = 0, 1, ⋯ , n - 1} . Let x, y ∈ θ^* be any two vertices of G. Then strength of connectedness between x and y is denoted by CONF_G (x, y) and defined by CONF_G (x, y) = ∨ {S (P) : Pisany (x, y) path} , where ∨ denotes maximum. For a path P, if S (P) is equal to CONF_G (x, y), then P is (x, y) strongest path. G is called connected if for any x, y ∈ θ^*, CONF_G (x, y) >0. G called tree if there exist a FSG S = (θ, ω) which is a tree and spans G, such that for any a, b not in S, there exists a path P between a and b in S such that ρ (a, b) < S (P). Also S is a unique maximum spanning tree (MST) of G. Strong edges and δ-edges in a FG are defined in the next definition. Let (x, y) be an edge of G. Then the edge (x, y) is called α-strong or α-strong edge (α-st) if ρ (x, y) > CONF_{G
_xy} (x, y). The edge (x, y) is called β-strong or β-strong edge (β-st) if ρ (x, y) = CONF_{G
_xy} (x, y). The edge (x, y) is said to be δ-edge if it is neither α-st nor α-st. (x, y) is called strong edge if it is not a δ-edge. A strong path is a path which does not contain any δ-edge. Geodesic (GDS) between x and y is the shortest strong path between x to y. Sum of membership values (MVs) of all edges in the GDS is called the weight of the GDS. A FG G is called α-saturated (α-sat) if each vertex contains at least one α-st edge in G. and β-saturated (β-sat) if each vertex contains at least one β-st edge in G. If it is both α-sat and β-sat then G is called saturated FG (sat-FG).

The vertices and edges for a saturated cycle are characterized in the next two theorems.

Theorem 2.1. [3] Let G = (θ, ρ) be a saturated cycle with n vertex. Then (i) n is an even number. (ii) α-st edge and β-st edge alternatively appears in G.

Theorem 2.2. [3] Let G = (θ, ρ) be a saturated cycle. Then each β-st edge has constant strength.

Isomorphism between two FGs are defined below.

Definition 2.1. [18] Two FGs G₁ = (θ₁, ρ₁) and G₂ = (θ₂, ρ₂) are called isomorphic if there exist a bijective map $h : θ_{1}^{*} \to θ_{2}^{*}$ with any $x, y \in θ_{1}^{*}, ρ_{2} (h (x), h (y)) = ρ_{1} (x, y)$ .

In 2020, Kalathian et al. [10] defined first ZI for a FG as follows:

Definition 2.2. [10] Let G = (θ, ρ) be a FG. Then first ZI of the FG G is denoted by M (G) and is defined by: $M (G) = \sum_{v \in θ^{*}} θ (v) d^{2} (v),$ where d (v) is the degree of the vertex v.

In 2021, the definition of the first ZI for a FG was modified by Islam and Pal [8].

Definition 2.3. [8] Suppose G = (θ, ρ) be a FG. Then first ZI of the FG G is indicated by ZF₁ (G) and is defined by: ${ZF}_{1} (G) = \sum_{v \in θ^{*}} [θ (v) d (v)]^{2} .$

In 2020, Kalathian et al. [10] defined second ZI for a FG as:

Definition 2.4. [10] Let G = (θ, ρ) be a FG. Then the second ZI of the FG G is denoted by ZF₂ (G) and is defined by: ${ZF}_{2} (G) = \sum_{uv \in ρ^{*}} θ (u) θ (v) d (u) d (v) .$

In 2021, Islam and Pal [9] introduced F-index for a FG.

Definition 2.5. [9] Suppose G = (θ, ρ) be a FG. Then F-index of the FG Γ is indicated by FF (G) and is defined by: $FF (G) = \sum_{v \in θ^{*}} [θ (v) d (v)]^{3} .$

Note that, those topological indices (First Zagreb index, Second Zagreb index, F-index) are defined depending on the degree of the vertices. So those indices are degree based topological indices and used to calculate π-electron energy of a conjugate system. In this paper, We studied another types of topological indices which are distance based topological indices more specifically for fuzzy graph this topological indices are strong distance based topological indices.

3 Hyper-wiener index of a fuzzy graph

In 1947 [35], Harold Wiener first introduced Wiener index (WI) for connected crisp graphs defined as: $\begin{matrix} WI (G) = \sum_{u, v \in V} d (u, v) . \end{matrix}$

Inspired by this, Randic [23] defined the HWI as: $\begin{matrix} HWI (G) = \frac{1}{2} \sum_{u, v \in V} [d (u, v) + d^{2} (u, v)] . \end{matrix}$

In [3] Binu et al. defined the WI for a connected FG as: $\begin{matrix} FWI (G) = \sum_{x, y \in θ^{*}} θ (x) θ (y) d_{S} (x, y), \end{matrix}$ where d_S (x, y) is called strong distance between x and y and which is minimum sum of weights of GDS between x to y. Now, a fuzzy hyper-Wiener index is defined which is a more generalization of WI, HWI, FWI and provides some interesting results on this index. Also an application of this index is provided in this article.

Definition 3.1. Let G = (θ, ρ) be a connected FG. The FHWI of G is defined by: $\begin{matrix} FHWI (G) = \frac{1}{2} \sum_{a, b \in θ^{*}} θ (a) θ (b) [d_{S} (a, b) + d_{S}^{2} (a, b)] . \end{matrix}$

Note that for each example of FGs, G = (θ, ρ) it is assumed that θ (u) =1 ∀ u ∈ θ^*.

In the next example, FHWI of the FG in Fig. 1 is calculated.

Fig. 1

A FG G and a PFSG H with FHWI (H) < FHWI (G).

Example 3.1. Let G = (θ, ρ) be the FG in Fig. 1 with θ^* = {a, ⋯ , e}; ρ (ab) =0.1, ρ (ac) =0.1, ρ (bc) =0.1, ρ (cd) =0.1, ρ (ce) =0.1, ρ (de) =0.1. Clearly all edges of G are strong. Then any shortest path between u, v are the GDS. Hence, $\begin{matrix} 2 FHWI (G) & = \sum_{a, b \in θ^{*}} θ (a) θ (b) [d_{S} (a, b) + d_{S}^{2} (a, b)] \\ = 6 (0.1 + 0 . 1^{2}) + 4 (0.2 + 0 . 2^{2}) \\ = 1.62 . \end{matrix}$ Therefore, FHWI (G) =0.81.

In the next example, FHWI of the PFSG H depicted in Fig. 1 is calculated and compared with original FG G.

Example 3.2. Let H = (θ, ω) be the PFSG of G = (θ, ρ) in Fig. 1 with ω (ce) =0, and ω (uv) = ρ (uv) ∀ (u, v) ∈ ρ^* \ {ce}. Then, $\begin{matrix} 2 FHWI (H) & = 5 (0.1 + 0 . 1^{2}) + 3 (0.2 + 0 . 2^{2}) \\ + 2 (0.3 + 0 . 3^{2}) \\ = 2.05 . \end{matrix}$ Therefore, FHWI (H) =1.025 > FHWI (G).

Now some questions are arisen about FHWI during edge deletion or edge addition in a FG.

(1) Suppose an edge is deleted in a connected FG such that the deleted fuzzy graph is also connected. Is FHWI decreased or increased or the same?

(2) Suppose an edge is added in a connected FG. Is FHWI decreased or increased or the same?

The answer of the questions are FHWI may be decreased or increased or unchanged when an edge is deleted or added in a FG. Clearly, FHWI is unchanged if an δ-edge is deleted or added in that fuzzy graph. The next example is about other queries.

Example 3.3. Let us consider a FG G = (θ, ρ) in Fig. 2 with θ^* = {a, ⋯ , l}; ρ (bc) =0.5, ρ (ab) =1, ρ (cd) =0.6, ρ (de) =0.4, ρ (ef) =0.1, ρ (af) =0.1, ρ (ae) =0.4, ρ (eg) =1, ρ (gh) =0.4, ρ (hi) =1, ρ (ij) =0.4, ρ (jk) =1, ρ (kl) =0.5, ρ (gl) =0.5, ρ (gk) =0.4 and θ (v) =1, ∀ v ∈ θ^*.

Fig. 2

A FG G with FHWI (G) < FHWI (G_ae) and FHWI (G) > FHWI (G_gk)

Strong distance path matrix D_P of G is defined as (i, j)th entry of D_P is: $\frac{1}{2} {d_{s} (v_{i}, v_{j}) + d_{s}^{2} (v_{i}, v_{j})} .$ Now the D_P (G) of Fig. 2 is given in Table 1.

Table 1

Strong distance path matrix D_p for the graph G shown in Fig. 2

	a	b	c	d	e	f	g	h	i	j	k	l
a	0	1	1.875	0.72	0.28	0.055	1.68	2.52	5.32	5.32	2.52	2.755
b	1	0	0.375	1.155	1.68	1.155	4.08	5.32	9.12	9.12	5.32	5.655
c	1.875	0.375	0	0.48	1	1.155	3	4.08	7.48	7.48	4.08	4.375
d	0.72	1.155	0.48	0	0.28	0.375	1.68	2.52	5.32	5.32	2.52	2.755
e	0.28	1.68	1	0.28	0	0.055	1	1.68	4.08	4.08	1.68	1.875
f	0.055	1.155	1.155	0.375	0.055	0	1.155	1.875	4.375	4.375	1.875	2.08
g	1.68	4.08	3	1.68	1	1.155	0	0.28	1.68	1.68	0.28	0.375
h	2.52	5.32	4.08	2.52	1.68	1.875	0.28	0	1	1.68	0.72	0.855
i	5.32	9.12	7.48	5.32	4.08	4.375	1.68	1	0	0.28	1.68	2.755
j	5.32	9.12	7.48	5.32	4.08	4.375	1.68	1.68	0.28	0	1	1.875
k	2.52	5.32	4.08	2.52	1.68	1.875	0.28	0.72	1.68	1	0	0.375
l	2.755	5.655	4.375	2.755	1.875	2.08	0.375	0.855	2.755	1.875	0.375	0

Table 2

Strong distance path matrix of G_ae

	a	b	c	d	e	f	g	h	i	j	k	l
a	0	1	1.875	0.48	0.12	0.055	1.32	2.08	4.68	4.68	2.08	2.295
b	1	0	0.375	1.155	1.32	1.155	3.52	4.68	8.28	8.28	4.68	4.995
c	1.875	0.375	0	0.48	1	1.155	3	4.08	7.48	7.48	4.08	4.375
d	0.48	1.155	0.48	0	0.28	0.375	1.68	2.52	5.32	5.32	2.52	2.755
e	0.12	1.32	1	0.28	0	0.055	1	1.68	4.08	4.08	1.68	1.875
f	0.055	1.155	1.155	0.375	0.055	0	1.155	1.875	4.375	4.375	1.875	2.08
g	1.32	3.52	3	1.68	1	1.155	0	0.28	1.68	1.68	0.28	0.375
h	2.08	4.68	4.08	2.52	1.68	1.875	0.28	0	1	1.68	0.72	0.855
i	4.68	8.28	7.48	5.32	4.08	4.375	1.68	1	0	0.28	1.68	2.755
j	4.68	8.28	7.48	5.32	4.08	4.375	1.68	1.68	0.28	0	1	1.875
k	2.08	4.68	4.08	2.52	1.68	1.875	0.28	0.72	1.68	1	0	0.375
l	2.295	4.995	4.375	2.755	1.875	2.08	0.375	0.855	2.755	1.875	0.375	0

Table 3

Strong distance path matrix of G_gk

	a	b	c	d	e	f	g	h	i	j	k	l
a	0	1	1.875	0.72	0.28	0.055	1.68	2.52	5.32	6.72	4.08	2.755
b	1	0	0.375	1.155	1.68	1.155	4.08	5.32	9.12	10.92	7.48	5.655
c	1.875	0.375	0	0.48	1	1.155	3	4.08	7.48	9.12	6	4.375
d	0.72	1.155	0.48	0	0.28	0.375	1.68	2.52	5.32	6.72	4.08	2.755
e	0.28	1.68	1	0.28	0	0.055	1	1.68	4.08	5.32	3	1.875
f	0.055	1.155	1.155	0.375	0.055	0	1.155	1.875	4.375	5.655	3.255	2.08
g	1.68	4.08	3	1.68	1	1.155	0	0.28	1.68	2.52	1	0.375
h	2.52	5.32	4.08	2.52	1.68	1.875	0.28	0	1	1.68	1.68	0.855
i	5.32	9.12	7.48	5.32	4.08	4.375	1.68	1	0	0.28	1.68	2.755
j	6.72	10.92	9.12	6.72	5.32	5.655	2.52	1.68	0.28	0	1	1.875
k	4.08	7.48	6	4.08	3	3.255	1	1.68	1.68	1	0	0.375
l	2.755	5.655	4.375	2.755	1.875	2.08	0.375	0.855	2.755	1.875	0.375	0

Since FHWI of a FG G is obtained by summing all upper triangular entries of D_P (G). So FHWI (G) =167.625. Now we consider the distance path matrix of G_ae (see Table 2). In this case, FHWI (G_ae) =159.705 < FHWI (G). The Table 3 is the distance path matrix of G_gk is provided. Here FHWI (G_gk) =188.805 > FHWI (G). Therefore, this example shows that deletion or addition of an edge does not imply FHWI increase or decrease.

In the next theorem, a bound for FHWI is provided for a connected FG.

Theorem 3.1. Let G = (θ, ρ) be a n-vertex connected FG. If t = ∧ {ρ (ab) |ab ∈ ρ^* and ab is a strong edge}, then, $FHWI (G) \geq \frac{t}{2} .$

Proof. As t = ∧ {ρ (ab) |ab ∈ ρ^* and ab is a strong edge}. Let a, b ∈ θ^*. If ab ∈ ρ^*, then d_S (a, b) ≥ t. There are |ρ^*| pairs of unordered vertex say (a, b) in G with d_S (a, b) ≥ t and $(\begin{matrix} n \\ 2 \end{matrix})$ - |ρ^*| pairs of unordered vertex (u, v) in G with d_S (u, v) ≥2t. Thus,

$\begin{matrix} 2 FHWI (G) & = \sum_{a, b \in θ^{*}} [d_{S} (a, b) + d_{S}^{2} (a, b)] \\ \geq | ρ^{*} | (t + t^{2}) + {(\begin{matrix} n \\ 2 \end{matrix}) - | ρ^{*} |} (2 t + 4 t^{2}) \\ = t [n (n - 1) (2 t + 1) - | ρ^{*} | (3 t + 1)] . \end{matrix}$

In the next theorem, FHWI is discussed for isomorphic FGs.

Theorem 3.2. Let G₁ = (θ₁, ρ₁) and G₂ = (θ₂, ρ₂) be isomorphic FGs. Then FHWI (G₁) = FHWI (G₂).

Proof. Let G₁ = (θ₁, ρ₁) and G₂ = (θ₂, ρ₂) are isomorphic FG. Let φ be the bijection from $θ_{1}^{*}$ to $θ_{2}^{*}$ such that θ₁ (a) = θ₂ (φ (a)) for $a \in θ_{1}^{*}$ and ρ₁ (ab) = ρ₂ (φ (a) φ (b)) for $ab \in ρ_{1}^{*}$ . For a, b ∈ θ^*, let P_a,b be the path which contributes d_S (a, b). For every edge uv ∈ P_a,b, there exist an edge φ (u) φ (v) in G₂ which satisfies ρ₁ (uv) = ρ₂ (φ (u) φ (v)) . Thus for every path P_a,b in G₁, there is a path P_φ(a),φ(b) in G₂ with minimum sum of MVs of edges of P for every shortest strong paths between φ (u) and φ (v). Hence d_S (a, b) = d_S (φ (a) , φ (b)) . Therefore,

$\begin{matrix} FHWI (G_{1}) = & \frac{1}{2} \sum_{a, b \in θ_{1}^{*}} θ_{1} (a) θ_{1} (b) [d_{S} (a, b) + d_{S}^{2} (a, b)] \\ = & \frac{1}{2} \sum_{φ (a), φ (b) \in θ_{2}^{*}} θ_{2} (φ (a)) θ_{2} (φ (b)) \\ [d_{S} (φ (a), φ (b)) + d_{S}^{2} (φ (a), φ (b))] \\ = & FHWI (G_{2}) . \end{matrix}$

In the next theorem, a bound of FHWI for a path is provided.

Theorem 3.3. Let P_n = P (u₁, u₂, ⋯ , u_n) be n-vertex path. Then, $FHWI (P_{n}) \leq \frac{1}{6} (n^{4} + n^{3} - 35 n^{2} + 67 n - 48) .$

Proof. Clearly, each edge of P_n is strong. Then

$\begin{matrix} 2 FHWI (P_{n}) & = ρ (e_{1}) + ρ^{2} (e_{1}) + ρ (e_{1}) + ρ (e_{2}) \\ + {ρ (e_{1}) + ρ (e_{2})}^{2} + \dots \\ + ρ (e_{2}) + ρ^{2} (e_{2}) + ρ (e_{2}) + ρ (e_{3}) \\ + {ρ (e_{2}) + ρ (e_{3})}^{2} + \dots \\ + ρ (e_{n - 1}) + ρ^{2} (e_{n - 1}) \\ = \sum_{i = 1}^{n - 1} i (n - i) {ρ (e_{i}) + ρ^{2} (e_{i})} \\ + 2 \sum_{i = 2}^{n - 1} 1 \cdot (n - i) ρ (e_{1}) ρ (e_{i}) + \dots \\ + 2 \sum_{i = 2}^{n - 1} k \cdot (n - i - k + 1) ρ (e_{k}) ρ (e_{i + k - 1}) \\ + \dots + 2 (n - 2) ρ (e_{n - 2}) ρ (e_{n - 1}) \\ = \sum_{i = 1}^{n - 1} i (n - i) {ρ (e_{i}) + ρ^{2} (e_{i})} \\ + 2 \sum_{j = 1}^{n - 2} \sum_{i = 2}^{n - 1} j (n - i - j + 1) ρ (e_{j}) ρ (e_{i + j - 1}) \\ ∴ FHWI (P_{n}) & \leq \sum_{i = 1}^{n - 1} i (n - i) + \sum_{j = 1}^{n - 2} \sum_{i = 2}^{n - 1} j (n - i - j + 1) \\ \leq n \sum_{i = 1}^{n - 1} i - \sum_{i = 1}^{n - 1} i^{2} + (n + 1) \sum_{j = 1}^{n - 2} \sum_{i = 2}^{n - 1} j \\ - \sum_{j = 1}^{n - 2} \sum_{i = 2}^{n - 1} j^{2} - \sum_{j = 1}^{n - 2} \sum_{i = 2}^{n - 1} ij \\ = \frac{1}{2} n^{2} (n - 1) - \frac{1}{6} n (n - 1) (2 n - 1) \\ + \frac{1}{2} (n^{2} - 1) (n - 2)^{2} - (n - 2)^{2} \\ - \frac{1}{6} (n - 2)^{2} (n - 1) (2 n - 3) \\ = \frac{1}{6} (n^{4} + n^{3} - 35 n^{2} + 67 n - 48) . \end{matrix}$ FHWI of a star is calculated in the next theorem.

Theorem 3.4. Let G = (θ, ρ) be a n-vertex star. Then, $FHWI (G) \leq \frac{1}{2} (n - 1) (n - 4) .$

Proof. Similar as Theorem 3.3, it follows $\begin{matrix} 2 FHWI = & \sum_{i = 1}^{n - 1} {ρ (e_{i}) + ρ^{2} (e_{i})} + \sum_{1 \leq i < j \leq n - 1} \\ [{ρ (e_{i}) + ρ (e_{j})} + {ρ (e_{i}) + ρ (e_{j})}^{2}] \\ = & (n - 1) \sum_{i = 1}^{n - 1} {ρ (e_{i}) + ρ^{2} (e_{i})} \\ + 2 \sum_{1 \leq i < j \leq n - 1} ρ (e_{i}) ρ (e_{j}) \\ ∴ FHWI \leq & \frac{n - 1}{2} \sum_{i = 1}^{n - 1} 2 + \sum_{1 \leq i < j \leq n - 1} 1 \\ = (n - 1)^{2} + \frac{1}{2} (n - 1) (n - 2) \\ = \frac{1}{2} (n - 1) (n - 4) . \end{matrix}$

In the next theorem, a relation between FHWI of a tree and its MST is established.

Theorem 3.5. Let S be the MST of the tree G = (θ, ρ). Then FHWI (G) = FHWI (S).

Proof. As S be the MST of the tree G, edges which are not in S are δ-edges. Note that any geodesic does not contain δ-edge. Hence the theorem proved.

In the next theorem, FHWI of a saturated cycle is studied.

Theorem 3.6. Let G = (θ, ρ) be a n-vertex saturated cycle. Let strength of the each α-st edge be α and each β-st edge be β, then $\begin{matrix} FHWI (G) \\ = {\begin{matrix} \frac{n^{3}}{32} (α + β) + \frac{n^{2} (n^{2} + 8)}{192} (α^{2} + β^{2}) \\ + \frac{n^{2} (n^{2} - 4)}{96} α β, n \equiv 0 (\mod 4) \\ \frac{n (n^{2} + 4)}{32} β + \frac{n (n^{2} - 4)}{32} α + \frac{n^{2} (n^{2} + 20)}{192} β^{2} \\ + \frac{n^{2} (n^{2} - 4)}{192} α^{2} + \frac{n^{2} (n^{2} - 4)}{96} α β, n \equiv 2 (\mod 4) . \end{matrix} \end{matrix}$

Proof. It is clear that alternate edges of G have MVs α and β and all edges of the FC G is strong. By Theorem 2.1, n be an even number. GDS in G can be $1, 2, \dots, \frac{n}{2}$ . For $1 \leq p \leq \frac{n}{2}$ , define P_p = {(c, d) ∈ θ^* × θ^*| lengthofthegeodesicbetween c and d isp}. Then the cardinality of $P_{\frac{n}{2}}$ is $\frac{n}{2}$ . Now for any $(c, d) \in P_{\frac{n}{2}}$ , $d_{s} (c, d) = {\begin{matrix} \frac{n}{4} (α + β) & if n \equiv 0 (\mod 4) \\ β + \frac{n - 2}{4} (α + β) & if n \equiv 2 (\mod 4) . \end{matrix}$ For any vertex c ∈ θ^*, there are exactly two vertices at distance p from c. Number of such pairs of vertices is 2n. Avoiding repetition, n number of distinct pairs (r, s) ∈ θ^* × θ^* where length of the GDS between r and s is p. For $1 \leq p < \frac{n}{2}, p$ even and let (c, d) ∈ P_p then $d_{s} (c, d) = \frac{p}{2} (α + β)$ . Let $1 \leq p < \frac{n}{2}$ , p odd and c ∈ θ^*. Let d, e ∈ θ^* such that the distance from c to d and e is p. Then one of the GDS of length p from c contains $\frac{p + 1}{2}$ α-st, $\frac{p - 1}{2}$ β-st edges and other geodesic contain $\frac{p - 1}{2}$ α-st, $\frac{p + 1}{2}$ β-st edges.

Then for n ≡ 0 (mod 4):

$\begin{matrix} 2 FHWI (G) & = \sum_{(c, d) \in P_{\frac{n}{2}}} [d_{s} (c, d) + d_{s}^{2} (c, d)] \\ + \sum_{p = 2, 4, \dots}^{\frac{n}{2} - 2} (\sum_{(c, d) \in P_{p}} [d_{s} (c, d) + d_{s}^{2} (c, d)]) \\ + \sum_{p = 1, 3, \dots}^{\frac{n}{2} - 1} (\sum_{(c, d) \in P_{p}} [d_{s} (c, d) + d_{s}^{2} (c, d)]) \\ = \frac{n}{2} [\frac{n}{4} (α + β) + \frac{n^{2}}{16} (α + β)^{2}] \\ + \sum_{p = 2, 4, \dots}^{\frac{n}{2} - 2} n [\frac{p}{2} (α + β) + \frac{p^{2}}{4} (α + β)^{2}] \\ + \sum_{p = 1, 3, \dots}^{\frac{n}{2} - 1} \frac{n}{2} [\frac{p + 1}{2} β + \frac{p - 1}{2} α \\ + (\frac{p + 1}{2} β + \frac{p - 1}{2} α)^{2} + \frac{p - 1}{2} β \\ + \frac{p + 1}{2} α + (\frac{p - 1}{2} β + \frac{p + 1}{2} α)^{2}] \\ = \frac{n}{2} [\frac{n}{4} (α + β) + \frac{n^{2}}{16} (α + β)^{2}] \\ + \sum_{p = 2, 4, \dots}^{\frac{n}{2} - 2} n [\frac{p}{2} (α + β) + \frac{p^{2}}{4} (α + β)^{2}] \\ + \sum_{p = 1, 3, \dots}^{\frac{n}{2} - 1} \frac{n}{2} [p (α + β) \\ + (α^{2} + β^{2}) + \frac{l^{2} - 1}{2} (α + β)^{2}] \\ = \frac{n^{3}}{16} (α + β) + \frac{n^{2}}{8} (α^{2} + β^{2}) \\ + \frac{n^{2} (n^{2} - 4)}{96} (α + β)^{2} . \end{matrix}$

Now for n ≡ 2 (mod 4): $\begin{matrix} 2 FHWI (G) & = \sum_{(c, d) \in P_{\frac{n}{2}}} [d_{s} (c, d) + d_{s}^{2} (c, d)] \\ + \sum_{p = 2, 4, \dots}^{\frac{n}{2} - 1} \sum_{(c, d) \in P_{p}} [d_{s} (c, d) + d_{s}^{2} (c, d)] \\ + \sum_{p = 1, 3, \dots}^{\frac{n}{2} - 2} \sum_{(c, d) \in P_{p}} [d_{s} (a, b) + d_{s}^{2} (a, b)] \end{matrix}$ $\begin{matrix} = \frac{n}{2} [β + \frac{n - 2}{4} (α + β) \\ + (β + \frac{n - 2}{4} (α + β))^{2}] \\ + \sum_{p = 2, 4, \dots}^{\frac{n}{2} - 1} n [\frac{p}{2} (α + β) + \frac{p^{2}}{4} (α + β)^{2}] \\ + \sum_{p = 1, 3, \dots}^{\frac{n}{2} - 2} \frac{n}{2} [\frac{p + 1}{2} β + \frac{p - 1}{2} α \\ + \frac{p - 1}{2} β + \frac{p + 1}{2} α \\ + (\frac{p + 1}{2} β + \frac{p - 1}{2} α)^{2} \\ + (\frac{p - 1}{2} β + \frac{p + 1}{2} α)^{2}] \\ = \frac{n (n^{2} + 4)}{16} β + \frac{n (n^{2} - 4)}{16} α \\ + \frac{n^{2} (n^{2} + 20)}{96} β^{2} \\ + \frac{n^{2} (n^{2} - 4)}{96} α^{2} + \frac{n^{2} (n^{2} - 4)}{48} α β . \end{matrix}$

4 Application of fuzzy hyper-Wiener index in share market

Shares are the most important factor of a company and one of the most significant phases in the business network. Investors take a vital risk but there are chances to earn huge benefits. In this paper, five interconnected companies A₁, A₂, A₃, A₄ and A₅ are chosen. Aim of this section is to find out which company is better for investment. Some parameters are needed to compare those companies. The significant parameters are:

P/E ratio of the company (p₁): The P/E ratio can help an investor to determine whether a stock is overvalued or under priced. This ratio is calculated by the formula: $P / E ratio = \frac{Current price of the stock}{{Company}^{'} s earning of the share} .$ High value of this ratio indicates investors are willing to pay a higher price.

Earning of the company (p₂): One way to evaluate companies to find good shares to buy is to look at its past earning and future earnings projections.

Quality of the management (p₃): Before investing in shares of a company, an investor should definitely look at the past performance of the company.

All those parameters are linguistic terms and the value of those parameters differ for different circumstances. To handle such a situation, we modelled the problem as a fuzzy graph.

Score values of the parameters for each company are provided in Table 4 and score values of the influenced relation between interconnected companies for each parameter are provided in Table 5. Note that all the score values are taken in [0,1]. Now the combined fuzzy graph depicted in Figure 3, for those parameters is constructed whose vertex set is the set of companies and vertex MV of a vertex is the score value of the parameter for corresponding company. There is an edge between two companies if they are interconnected and the score value of the influenced relation between those companies represents the edge membership value. Now we consider the strong distance matrix of the fuzzy graph whose (i, j)th entry is defined as d_s (v_i, v_j). Now strong distance matrix of G, G_{A
₁}, G_{A
₂}, G_{A
₃}, G_{A
₄}, G_{A
₅} is given in Table 6-11. Then FHWI of G for those parameter p₁, p₂, p₃ is calculated by the formula $\frac{1}{2} \sum_{a, b \in θ^{*}} θ (a) θ (b) [d_{S} (a, b) + d_{S}^{2} (a, b)]$ and the values are 3.0332, 2.64575, 2.5859 respectively. TFHWI of a graph is the sum of FHWI of the graph for each parameter. Decision score of a company A is calculated by the formula $\frac{TFHWI (G) - TFHWI (G_{A})}{TFHWI (G)} .$ The FHWI, TFHWI for each parameter of the graph G_{A
₁}, G_{A
₂}, G_{A
₃}, G_{A
₄}, G_{A
₅} and Score of the companies are listed in the Table 12. As the decision score of a company is directly dependent on the parameters and when one of the parameters is increased then the decision score of the company is also increased so higher decision score indicates that company is better to invest for investors. A₂ is the best company among those companies as the company A₂ has the highest decision score among them.

Table 4
Score value of the parameters for each company

p₁ p₂ p₃

A ₁ 0.7 0.8 0.6

A ₂ 0.8 0.5 0.8

A ₃ 0.6 0.7 0.7

A ₄ 0.6 0.8 0.7

A ₅ 0.7 0.6 0.5

	p₁	p₂	p₃
A ₁	0.7	0.8	0.6
A ₂	0.8	0.5	0.8
A ₃	0.6	0.7	0.7
A ₄	0.6	0.8	0.7
A ₅	0.7	0.6	0.5

Table 5

Score value of the influenced relation between interconnected companies for each parameters

p ₁	A ₁	A ₂	A ₃	A ₄	A ₅	p₂	A ₁	A ₂	A ₃	A ₄	A ₅	p₃	A ₁	A ₂	A ₃	A ₄	A ₅
A ₁		0.7	0.6	0.5	0.6	A ₁		0.5	0.7	0.6	0.6	A ₁		0.6	0.5	0.5	0.5
A ₂	0.7		0.6		0.6	A ₂	0.5		0.5		0.5	A ₂	0.6		0.7		0.5
A ₃	0.6	0.6		0.6		A ₃	0.7	0.5		0.6		A ₃	0.5	0.7		0.7
A ₄		0.5		0.6		0.6	A ₄	0.6		0.6		0.6	A ₄	0.5		0.7		0.5
A ₅	0.6	0.6		0.6		A ₅	0.6	0.5		0.6		A ₅	0.5	0.5		0.6

Fig. 3

Combined fuzzy graph representation of the problem.

Table 6

Strong distance matrix of G

G	A ₁	A ₂	A ₃	A ₄	A ₅
A ₁		(0.7,0.5,0.6)	(0.6,0.7,0.5)	(0.6,0.6,0.5)	(0.6,0.6,0.5)
A ₂	(0.7,0.5,0.6)		(0.6,0.5,0.7)	(1.2,1.1,1.0)	(0.6,0.5,0.5)
A ₃	(0.6,0.7,0.5)	(0.6,0.5,0.7)		(0.6,0.6,0.7)	(1.2,1.0,1.2)
A ₄	(0.6,0.6,0.5)	(1.2,1.1,1.0)	(0.6,0.6,0.7)		(0.6,0.6,0.5)
A ₅	(0.6,0.6,0.5)	(0.6,0.5,0.5)	(1.2,1.0,1.2)	(0.6,0.6,0.5)

Table 7

Strong distance matrix of G_{A
₁} .

G _{A ₁}	A ₂	A ₃	A ₄	A ₅
A ₂		(0.6,0.5,0.7)	(1.2,1.1,1.0)	(0.6,0.5,0.5)
A ₃	(0.6,0.5,0.7)		(0.6,0.6,0.7)	(1.2,1.0,1.2)
A ₄	(1.2,1.1,1.0)	(0.6,0.6,0.7)		(0.6,0.6,0.5)
A ₅	(0.6,0.5,0.5)	(1.2,1.0,1.2)	(0.6,0.6,0.5)

Table 8

Strong distance matrix of G_{A
₂}

G _{A ₂}	A ₁	A ₃	A ₄	A ₅
A ₁		(0.6,0.7,0.5)	(1.2,0.6,0.5)	(0.6,0.6,0.5)
A ₃	(0.6,0.7,0.5)		(0.6,0.6,0.7)	(1.2,1.2,1.0)
A ₄	(1.2,0.6,0.5)	(0.6,0.6,0.7)		(0.6,0.6,0.5)
A ₅	(0.6,0.6,0.5)	(1.2,1.2,1.0)	(0.6,0.6,0.5)

Table 9

Strong distance matrix of G_{A
₃}

G _{A ₃}	A ₁	A ₂	A ₄	A ₅
A ₁		(0.7,0.5,0.6)	(1.2,0.6,0.5)	(0.6,0.6,0.5)
A ₂	(0.7,0.5,0.6)		(1.2,1.1,1.0)	(0.6,0.5,0.5)
A ₄	(1.2,0.6,0.5)	(1.2,1.1,1.0)		(0.6,0.5,0.5)
A ₅	(0.6,0.6,0.5)	(0.6,0.5,0.5)	(0.6,0.5,0.5)

Table 10

Strong distance matrix of G_{A
₄}

G _{A ₄}	A ₁	A ₂	A ₃	A ₅
A ₁		(0.7,0.5,0.6)	(0.6,0.7,1.3)	(0.6,0.6,0.5)
A ₂	(0.7,0.5,0.6)		(0.6,0.5,0.7)	(0.6,0.5,0.5)
A ₃	(0.6,0.7,1.3)	(0.6,0.5,0.7)		(1.2,1.0,1.2)
A ₅	(0.6,0.6,0.5)	(0.6,0.5,0.5)	(1.2,1.0,1.2)

Table 11

Strong distance matrix of G_{A
₅}

G _{A ₅}	A ₁	A ₂	A ₃	A ₄
A ₁		(0.7,0.5,0.6)	(0.6,0.7,1.3)	(1.2,0.6,2.0)
A ₂	(0.7,0.5,0.6)		(0.6,0.5,0.7)	(1.2,1.1,1.4)
A ₃	(0.6,0.7,1.3)	(0.6,0.5,0.7)		(0.6,0.6,0.7)
A ₄	(1.2,0.6,2.0)	(1.2,1.1,1.4)	(0.6,0.6,0.7)

Table 12

Fuzzy hyper-Wiener index

Graph	Fuzzy hyper-Wiener index for p₁	Fuzzy hyper-Wiener index for p₂	Fuzzy hyper-Wiener index for p₃	Total Fuzzy hyper-Wiener index	Name of the company	Decision score of the company
G _{A ₁}	2.0616	1.62495	1.928	5.61455	A ₁	0.320671277
G _{A ₂}	1.92	1.9244	1.2003	5.0447	A ₂	0.389619896
G _{A ₃}	2.2268	1.4925	1.34165	5.06095	A ₃	0.387653738
G _{A ₄}	1.8236	1.37735	1.916	5.11695	A ₄	0.380878056
G _{A ₅}	2.126	1.65245	3.68385	7.4623	A ₅	0.097104001

5 Conclusion

TIs have an important role in spectral graph theory, chemical graph theory, biochemistry, etc. Most of the topological indices are defined in a crisp graph. As fuzzy graphs are more generalization of crisp graphs, those indices have more application in fuzzy graphs also. In this article, we introduced the fuzzy hyper-Wiener index (FHWI) and studied this index for various fuzzy graphs like path, cycle, star, etc and provided some interesting bounds of FHWI for those fuzzy graph. A lower bound of FHWI is established for n-vertex connected fuzzy graph depending on strength of a strong edges. A relation between FHWI of a tree and its maximum spanning tree is established and this index is calculated for a saturated cycle. Also, at the end of the article, an application in the share market of this index is presented.

The limitations and future scopes of the study are:

(i) Good lower bound of the FHWI for n-vertex connected FG is presented here but we cannot provide the good upper bound.

(ii) Good upper bound of the FHWI for path and star are presented here but we cannot provide the good lower bound.

(iii) Which n-vertex tree (fuzzy) has the maximum FHWI?

(iv) Which n-vertex tree (fuzzy) has the minimum FHWI?

(v) Which n-vertex connected FG has the maximum FHWI?

(vi) Which n-vertex connected FG has the minimum FHWI?

Footnotes

Acknowledgments

The authors are highly thankful to the honorable Editor in Chief and anonymous reviewers for their valuable suggestions which significantly improved the quality and representation of the paper. The first author is thankful to the University Grant Commission (UGC) Govt. of India for financial support under UGC-Ref. No.: 1144/ (CSIR-UGC NET JUNE 2017) dated 26/12/2018.

Compliance with ethical standards

Disclosure statement. No potential conflict of interest was reported by the authors.

Ethical approval. This article does not contain any studies with human participants or animals performed by any of the authors.

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Hyper-Wiener index for fuzzy graph and its application in share market

Abstract

Keywords

Abbreviations

1 Introduction

1.1 Research background

1.2 Objective of the work

1.3 Framework of the article

2 Preliminaries

3 Hyper-wiener index of a fuzzy graph

Table 4 Score value of the parameters for each company p1 p2 p3 A 1 0.7 0.8 0.6 A 2 0.8 0.5 0.8 A 3 0.6 0.7 0.7 A 4 0.6 0.8 0.7 A 5 0.7 0.6 0.5

Footnotes

Acknowledgments

References

Table 4
Score value of the parameters for each company

p₁ p₂ p₃

A ₁ 0.7 0.8 0.6

A ₂ 0.8 0.5 0.8

A ₃ 0.6 0.7 0.7

A ₄ 0.6 0.8 0.7

A ₅ 0.7 0.6 0.5