Some topological indices in fuzzy graphs

Abstract

A fuzzy graph is one of the versatile application tools in the field of mathematics, which allows the user to easily describe the fuzzy relation between any objects. The nature of fuzziness is favorable for any environment, which supports to predict the problem and solving it. Fuzzy graphs are beneficial to give more precision and flexibility to the system as compared to the classical model (i.e.,) crisp theory. A topological index is a numerical quantity for the structural graph of the molecule and it can be represented through Graph theory. Moreover, its application not only in the field of chemistry can also be applied in areas including computer science, networking, etc. A lot of topological indices are available in chemical-graph theory and H. Wiener proposed the first index to estimate the boiling point of alkanes called ‘Wiener index’. Many topological indices exist only in the crisp but it’s new to the fuzzy graph environment. The main aim of this paper is to define the topological indices in fuzzy graphs. Here, indices defined in fuzzy graphs are Modified Wiener index, Hyper Wiener index, Schultz index, Gutman index, Zagreb indices, Harmonic index, and Randić index with illustrations. Bounds for some of the indices are proved. The algorithms for distance matrix and MWI are shown. Finally, the application of these indices is discussed.

Keywords

Fuzzy graphs distance matrix topological indices operations on topological indices

1 Introduction

Generally, graphs are user-friendly by allowing one can transform the information between the desired objects through connections. Therefore, it’s convenient to express any model as graph structure which helps to manipulate the nature of the problem can be made into an advantageous one for future resources. A graph consists of two important collections namely vertex set and edge set. Objects present in the graph are referred to as vertices. These vertices are connected by an arc or line that is said to be an edge. During critical situations, there is a chance that any system can never be performed 100% efficient. Likewise, there is an opportunity of uncertainty in graph notion’s (i.e.), vertices and edges have fuzziness, and it’s necessary to define the field of fuzzy graphs.

Graph theory concepts are widely used in areas including science and engineering such as: designing communication networks, VLSI circuits, electric circuits, etc. Vagueness in the network or circuit can lead to study the graph structure in fuzzy. Fuzzy graph theory deals with the uncertainty of vertices and edges call as membership values. Interestingly, these membership values of the fuzzy graph look different from the crisp approach. Initially, the concept of fuzzy graphs was first introduced by A. Rosenfeld; it becomes a vast area of research due to its many applications in fields such as neural networks, artificial intelligence, expert systems, database theory, pattern recognition, and computer networks. Many graph theory notions are expressed in fuzzy graphs more specifically; planar graph, coloring, labeling, etc. have more important results in fuzzy graphs.

In chemical science, a topological representation of the molecule is called the molecular graph. It is a collection of nodes representing the atoms in the molecule and the chemical bonds considered as edges. A topological index is the graph invariant number calculated from the molecular graph. The Wiener number or Wiener index is the first topological index later, many indices are developed due to its application in chemistry such as Zagreb indices, Randić, Harmonic and Gutman indices, etc. are some of them. In [18], J.N. Moderson et al. developed the Wiener index in fuzzy graph such that every membership value of elements is important and compared with the fuzzy connectivity index. This can be applied over real-life fields such as illegal immigration [18], trafficking [18], and routing issues [18]. The main aim of the paper is to develop the new form of definitions for some of the topological indices in the fuzzy graph which are present in the chemical-graph theory.

The paper organized as follows: In Section 2, the literature review of the present study is discussed. Necessary definition in graphs, topological indices, and fuzzy graphs are shown as related works in Section 3. In Section 4, initiated with the definition of Modified Wiener index followed with Hyper Wiener, Schultz, Gutman, and Planar indices are defined with example. Some of the important theorems on boundedness are proved in it. Another important part of topological indices are Zagreb indices, Randić and Harmonic indices are defined with example in Section 5 and proved a relation between the operation on topological indices in fuzzy graphs. Algorithms for distance matrix and MWI are shown in Section 5. Finally, Section 6 & 7 provides application and conclusion with future works following references.

2 Previous works

The basics of graph theory were noted in [9, 29]. Graph theory plays a major role in Science and Engineering, for instance, physical-graph theory [3], chemical-graph theory [7, 27], etc. H. Wiener (a chemist), who introduced the Wiener index in 1947 [30] and importance of indices are shown [1 , 28]. Usage of Wiener indices is to find the properties of alkanes that are known to be paraffin. After Wiener index, many topological indices were developed such as Hyper-Wiener [12], Schultz [8], Gutman [8], Zagreb [11, 15], Harmonic [14], Randic [4], etc.

L. A. Zadeh introduced fuzzy sets in his seminal paper, in 1965 [32]. Many research works were carried out in this area for the solution to practical research problems due to its solving nature. Graph theory with fuzzy sets forms an interesting tool known as the ‘Fuzzy graph’ coined by A. Rosenfeld in 1975 [20]. Special topics in graphs are known to be planar, labeling, coloring was transliterated into the fuzzy graph. [17–19 , 26]. The first application of a fuzzy graph in chemical sciences was introduced by J. Xu in 1997 [31]. Recently, the properties in topological indices are shown in [2 , 25]. Topological indices such as Wiener and Connectivity index in fuzzy graphs were introduced by J.N. Moderson et. al in 2019 [18].

3 Preliminaries

3.1 Related works

Here some basic definitions from graphs are demonstrated using undirected graphs [9]. A graph is an ordered pair G = (V, E), where V is the set of vertices and E is the set of edges in G. A subgraph of a graph G = (V, E) is a graph H = (V, F), where W ⊆ V and F ⊆ E. A simple graph is an undirected graph that has no loops. Two vertices x and y in G are said to be adjacent in G if (x, y) is an edge of G.

The definitions of topological indices in graph theory are shown.

Definition 1 [30]. The Wiener index (W (G)) of G is defined by $W (G) = \sum_{u, v \in V (G)} d_{G} (u, v)$ where d (u, v) denotes the shortest distance between the vertices u and v.

Definition 2 [12]. The Hyper-Wiener index (WW (G)) is defined as $WW (G) = \frac{1}{2} (\sum d_{G} {(u, v)}^{2} + \sum d_{G} (u, v))$

Definition 3 [8]. The Schultz index (Sc (G)) is defined as: $Sc (G) = \frac{1}{2} \sum_{u, v \in V} (d (u) + d (v)) (d_{G} (u, v))$

Definition 4 [8]. The Gutman index (Gut (G)) is defined as $Gut (G) = \sum_{u, v \in V} d (u) d (v) (d_{G} (u, v))$

Definition 5 [11]. The first Zagreb index M₁ (G) and second Zagreb index M₂ (G) is defined as follows: $M_{1} (G) = \sum_{u_{i} \in V} d_{i}^{2} {andM}_{2} (G) = \sum_{u_{i} u_{j} \in E} d_{i} d_{j}$

Definition 6 [4]. The Randič index (R (G)) is defined as $R (G) = \sum_{uv \in E} \frac{1}{\sqrt{d (u) d (v)}}$

Definition 7 [14]. The Harmonic index (H (G)) is defined as $H (G) = \sum_{uv \in E} \frac{2}{d (u) + d (v)}$

A fuzzy graph is a pair of functions such that denoted by G : (V, σ, μ), where σ is a fuzzy subset of a set V and μ is a fuzzy relation on σ. It is assumed that V is finite and nonempty, and μ is reflexive and symmetric. Thus, if G : (V, σ, μ) is a fuzzy graph, then σ : V → [0, 1], and μ : V × V → [0, 1] is such that μ (u, v) ≤ σ (u) ∧ σ (v) , ∀ u, v ∈ V, where ∧ denotes the minimum.

The order and size of the fuzzy graph are denoted by m and n respectively. Thus m = ∑σ (u) and n = ∑μ (u, v). The degree of a vertex u in G is defined by d_G (u) = ∑_(u,v)∈V×Vμ (u, v).

The union of two fuzzy graphs G₁ = (V₁, σ₁, μ₁) and G₂ = (V₂, σ₂, μ₂) is defined as a fuzzy graph G = G₁ ∪ G₂ where the underlying crisp graphs for $G_{1}^{*} = (V_{1}, E_{1})$ and $G_{1}^{*} = (V_{2}, E_{2})$ respectively. Then the union of the two graphs is $(σ_{1} \cup σ_{2}) (u) = {\begin{matrix} σ_{1} (u) & if u \in V_{1} - V_{2} \\ σ_{2} (u) & if u \in V_{2} - V_{1} \\ σ_{1} (u) \lor σ_{2} (u) & if u \in V_{1} \cap V_{2} \end{matrix}$ $\begin{matrix} (μ_{1} \cup μ_{2}) (u, v) \\ = {\begin{matrix} μ_{1} (u, v) & if μ_{1} (u, v) \in E_{1} - E_{2} \\ μ_{2} (u, v) & if μ_{2} (u, v) \in E_{2} - E_{1} \\ μ_{1} (u, v) \lor μ_{2} (u, v) & if μ (u, v) \in E_{1} \cap E_{2} \end{matrix} \end{matrix}$

The intersection of two fuzzy graphs G₁ = (V₁, σ₁, μ₁) and G₂ = (V₂, σ₂, μ₂) is defined as a fuzzy graph G = G₁ ∩ G₂ where the underlying crisp graphs for $G_{1}^{*} = (V_{1}, E_{1})$ and $G_{1}^{*} = (V_{2}, E_{2})$ respectively. Then the intersection of the two graphs is $(σ_{1} \cap σ_{2}) (u) = {\begin{matrix} 0 & if u \in V_{1} - V_{2} \\ 0 & if u \in V_{2} - V_{1} \\ σ_{1} (u) \land σ_{2} (u) & if u \in V_{1} \cap V_{2} \end{matrix}$ $\begin{matrix} (μ_{1} \cap μ_{2}) (u, v) \\ = {\begin{matrix} 0 & if μ_{1} (u, v) \in E_{1} - E_{2} \\ 0 & if μ_{2} (u, v) \in E_{2} - E_{1} \\ μ_{1} (u, v) \land μ_{2} (u, v) & if μ (u, v) \in E_{1} \cap E_{2} \end{matrix} \end{matrix}$

A fuzzy edge [16] can be classified into three types based on the strength call it as α-strong, β- strong, and δ- arc. An edge μ (x, y) in G is called α-strong if μ (x, y) > CONN_G-μ(x,y) (x, y). An edge μ (x, y) in G is called β-strong if μ (x, y) = CONN_G-μ(x,y) (x, y). An arc μ (x, y) in G is called δ- arc if μ (x, y) < CONN_G-μ(x,y) (x, y).

The following two definitions were given by J.N. Moderson et al. and it helps to construct this whole paper.

Definition 9 [18]. The Connectivity index of a fuzzy graph is defined as $CI (G) = \sum_{u, v \in σ} σ (u) σ (v) {CONN}_{G} (u, v)$

Definition 10 [18]. The Wiener index of a fuzzy graph is defined as $WI (G) = \sum_{u, v \in σ} σ (u) σ (v) d_{s} (u, v)$

3.2 Methodology

In the Advanced Fuzzy graph [18], the Wiener index in the fuzzy graph was introduced, then compared with the connectivity index and completely studied with some real-time applications. In this paper, a slight modification in the Wiener index formulae and calls it as Modified Wiener index which coincides with the results of the crisp approach. Moreover, topological indices such as the Hyper Wiener index, Schultz index, Gutman index, Planarity index, Zagreb indices, Randic index, and Harmonic index are defined with the help of the Wiener index.

4 Topological indices in fuzzy graphs – Part 1

Here the base of the topological indices is distance matrix discussed initially and this notion is important for constructing the topological indices.

4.1 Distance matrix

The distance matrix of the fuzzy graph G is that for every pair of vertices is the sum of the shortest path of the membership value of the edges between them. The distance matrix of the fuzzy graph is denoted by M_dst.

Example 1. For the fuzzy graph G as shown in Fig. 1, the distance matrix is obtained by $M_{dst} (G) = \begin{matrix} a & b & c \\ a & 0 & 0.2 & 0.1 \\ b & 0.2 & 0 & 0.1 \\ c & 0.1 & 0.1 & 0 \end{matrix}$

Fig. 1

Fuzzy graph for distance matrix.

4.2 Modified Wiener index of fuzzy graph

Wiener index is the first index in graph theory as well as fuzzy graphs. A slight modification in the Wiener index in the fuzzy graph and which coincides with the definition of graph theory.

Definition 2. The Modified Wiener index MWI (G) of a fuzzy graph G is defined as $MWI (G) = \frac{1}{2} \sum_{u, v \in σ} σ (u) σ (v) d_{s} (u, v)$ where d_s (u, v) is the distance between the membership value of the edges.

Example 3. Consider the fuzzy graph G = (V, σ, μ) with σ = {a, b, c, d, e} =1, μ (a, b) =0.5, μ (a, c) =0.8, μ (a, e) =0.3, μ (b, d) =0.28, μ (c, d) =0.75, μ (d, e) =0.4 and shown in Fig. 2.

Fig. 2

Fuzzy graph with MWI (G) =6.14.

To find the shortest path such that the distance between the vertices having minimum value.

Therefore, $d_{s} (a, b) = 0.5; d_{s} (a, c) = 0.8; d_{s} (a, d) = 0.7;$ $d_{s} (a, e) = 0.3; d_{s} (b, c) = 1.03; d_{s} (b, d) = 0.28;$ $d_{s} (b, e) = 0.68; d_{s} (c, d) = 0.75; d_{s} (c, e) = 1.1;$ and d_s (d, e) = 0.4. $M_{dst} (G) = \begin{matrix} a & b & c & d & e \\ a & 0 & 0.50 & 0.80 & 0.70 & 0.30 \\ b & 0.50 & 0 & 1.03 & 0.28 & 0.68 \\ c & 0.80 & 1.03 & 0 & 0.75 & 1.10 \\ d & 0.70 & 0.28 & 0.75 & 0 & 0.40 \\ e & 0.30 & 0.68 & 1.10 & 0.40 & 0 \end{matrix}$ which implies that $\sum_{u, v \in σ} σ (u) σ (v) d_{s} (u, v) = 13.08$ Hence, $MWI (G) = \frac{1}{2} \sum_{u, v \in σ} σ (u) σ (v) d_{s} (u, v) = 6.54$

4.2.1 The relation between the strength of edges in connected and disconnected graphs for MWI

Consider the fuzzy graph G = (V, σ, μ) and suppose the elimination of the membership value of the fuzzy edge increases the indices value. Since there exist three types of fuzzy edges (i.e.) α- strong, β- strong and δ- arc.

In Example 3, the fuzzy graph has α-strong edges and δ-arcs. Therefore, μ (a, b) , μ (a, c) , μ (c, d) and μ (d, e) are α- strong edges and μ (a, e) , μ (b, d) are δ - arcs. Therefore, the deletion of δ- arc (i.e., μ (a, e)) whose MWI value is greater than the deletion of α- strong edge (i.e. μ (a, b)) MWI value shown in Fig. 3(a) & 3(b). Hence, $\begin{matrix} MWI (G) = 6.54 & < MWI (G ∖ α - strongedge) = 6.99 \\ < MWI (G ∖ δ - arc) = 7.5 \end{matrix}$

In general, for every fuzzy graph whose elimination of edges in connected graph is $\begin{matrix} MWI (G) & < MWI (G ∖ α - strongedge) \\ < MWI (G ∖ β - strongedge) \\ < MWI (G ∖ δ - arc) \end{matrix}$

Suppose the elimination of fuzzy edge in G disjoints the graph and hence its disconnected. Consider the fuzzy graph G as shown in Fig. 3. The removal of any type has lesser value than the MWI (G). Therefore, MWI (G ∖ μ (x, u)) < MWI (G).

Fig. 3

(a) Deletion of α - strong (μ (a, b)) (b) Deletion of δ - arc (δ (a, b)) (c) Disconnected graph.

Example 4. Consider the Example 3, the subgraph of fuzzy graph G is obtained by deletion of fuzzy edges i.e. deletion of {μ (a, e) , μ (e, d)} in Fig. 3(c).

Therefore, $\begin{matrix} d_{s} (a, b) = 0.5; d_{s} (a, c) = 0.8; d_{s} (a, d) = 0.78 \\ d_{s} (b, c) = 1.03; d_{s} (b, d) = 0.28; d_{s} (c, d) = 0.75 \\ MWI (G - {μ (a, e), μ (e, d)}) \\ = \frac{1}{2} (\sum_{u, v \in σ andG ∖ {μ (a, e), {μ (e, d)}} σ (u) σ (v) d_{s} (u, v)) \\ = 4.14 < 6.54 = MWI (G) \end{matrix}$

4.2.2 Boundedness for modified wiener index

The upper and lower bounds for the Modified Wiener index are discussed with the condition the membership value of vertices to be in [0, 1] and the membership value of the edge to be in (0, 1).

Theorem 5. Let G be a fuzzy graph with m and n are order and size respectively. Then $MWI (G) \geq mnp, if | m^{*} | < | n^{*} |$ (1) $MWI (G) \geq \frac{1}{2} (mnp), if | m^{*} | \geq | n^{*} |$ (2) Where |m^*|, |n^*| are the number of strong and weak edges (δ- arcs) present in the fuzzy graph G respectively and p be the least membership value of the fuzzy edge in G.

Proof. Case(i): |m^*| < |n^*|

This theorem is proved by the method of induction on fuzzy graphs. For instance, let G be the fuzzy graph with each membership value of vertex taken as constant value 1. Then the membership value of edges are lies in (0, 1). Assume the fuzzy graph K₂, whose membership value of vertices and edges to be 1 and r ∈ (0, 1) respectively. Then $MWI (G) = \frac{1}{2} \sum_{u, v \in σ} σ (u) σ (v) d_{s} (u, v) = \frac{1}{2} (2 r) = r$

Therefore, mnp = 2r² and r ≥ 2r², only if r ∈ [0, 1). In K₂, presence one edge considered as α- strong and hence the case(i) is true.

Consider a fuzzy graph with a finite number of vertices and edges such that whose membership values is in [0, 1) and (0, 1) respectively. Let {a₁, a₂, a₃, … } and {e₁, e₂, e₃, … } be the vertices and edges of G = K_n-1 is true. To show that for the fuzzy graph G = K_n is true for MWI (G). Therefore, the order and size of G is determined as m = ∑a_i and n = ∑e_i. Consider {a₁, a₂, a₃, …, a_n } = 1 and {e₁, e₂, e₃, …, e_n } = r, r ∈ (0, 1) in K_n. Then Equation (3) is true for K_n-1 with m = N - 1 and $n = \frac{r \times (N - 2) (N - 1)}{2}$ where $\frac{(N - 2) (N - 1)}{2}$ be the number of fuzzy edges in K_n-1 and p = r. Since $MWI (G) + \frac{1}{2} (3 N^{2} - 5 N + 2) \geq 0$

$\begin{matrix} \frac{1}{2} \sum_{u, v \in σ} σ (u) σ (v) d_{s} (u, v) \\ \geq \frac{N - 1 \times r^{2} \times (N - 2) (N - 1)}{2} \end{matrix}$ (3) $\begin{matrix} \geq \frac{N - 1 \times r^{2} \times (N - 2) (N - 1)}{2} \\ - \frac{1}{2} (3 N^{2} - 5 N + 2) \end{matrix}$ $= \frac{N \times r^{2} \times N (N - 1)}{2} = MWI (G = K_{n})$ (4) $\frac{1}{2} \sum_{u, v \in σ^{*}} σ (u) σ (v) d_{s} (u, v) \geq mnp$

Hence, by Equation (4), the fuzzy graph G = K_n is true. Therefore, MWI (G) ≥ mnp.

Case(ii):|m^*| ≥ |n^*|

By using induction hypothesis, consider a path P₂ of fuzzy graph such that each vertex and edge has membership value as 1. Hence to find the lower bound for the Modified Wiener Index by using the Eq (2). Because, there is no weak edges in P₂ and therefore $MWI (G) = \frac{1}{2} \sum_{u, v \in σ} σ (u) σ (v) d_{s} (u, v) = \frac{1}{2} (2 r) = r$ $\frac{1}{2} (mnp) = \frac{1}{2} (2 \times r \times r) = r^{2}$

Hence, the result holds good for the fuzzy path P₂ and therefore $MWI (G) \geq \frac{1}{2} (mnp)$ . Assume the result is true for P_n-1.

Consider a fuzzy path P_n in the fuzzy graph such that the number of edges present in the fuzzy graph has a large number of strong edges compare to δ-arcs. Then, $\frac{1}{2} (mnp) = \frac{1}{2} (n . (r_{1} + r_{2} \dots + r_{n - 1}) . r_{j})$

And hence $MWI (G) > \frac{1}{2} (mnp)$

Hence the Equations (1) & (2) is true for the case n.

Theorem 6. Let G be a fuzzy graph with m and n are order and size respectively. Then $MWI (G) \leq mn$ (4)

Proof. Let G be the fuzzy graph such that m and n are its order and size respectively. To show that the upper bound of MWI (G) is mn. Since $2 \sum_{u, v \in σ} σ (u) σ (v) d_{s} (u, v) \leq 2 m (r_{1} + r_{2} \dots r_{n}) + 2 m$ (5) $\sum_{u, v \in σ} σ (u) σ (v) d_{s} (u, v) \leq 2 m$ (6)

On subtracting (5) - (6), we get $\begin{matrix} \sum_{u, v \in σ} σ (u) σ (v) d_{s} (u, v) \leq 2 mn + 2 m - 2 m \\ = 2 mn \\ \frac{1}{2} \sum_{u, v \in σ} σ (u) σ (v) d_{s} (u, v) \leq mn \\ MWI (G) \leq mn \end{matrix}$

Hence the theorem is proved.

Example 7. For the fuzzy graph in Example 3, we easily find out the upper and lower bound of the fuzzy graphs by using the Theorems 5 and 6. Therefore, the lower bound for the given fuzzy graph G is 4.5 and the upper bound is 15.15.

4.2.3 Disconnectedness in MWI

Suppose G be the connected fuzzy graphs containing all types of edges (i.e.,) strong or δ-edges. Then by removing the fuzzy edges so the given fuzzy graph becomes disconnected. Then the following result holds good for this type of condition when performed.

Theorem 8. Let G be the connected fuzzy graph. If G is disconnected when μ (x, y) is eliminated. Then the Modified Wiener index of two graphs should be MWI (G′) < MWI (G″) where the order m′ < m″.

Proof. Let G be the connected fuzzy graph and consider the fuzzy edge in G such that the elimination of the edge makes the fuzzy graph disconnected. Elimination of the fuzzy edge gives the fuzzy graph disconnected, then two new fuzzy graphs formed as G′ and G″ of G. Since the two new fuzzy graphs are subgraphs of G such that MWI (G′) < MWI (G) and MWI (G″) < MWI (G). If the elimination of fuzzy edge is αorβ strong or weak edge, it then the subgraph is always lesser its value than MWI (G). Hence the theorem is proved.

Corollary 9. Suppose consider a fuzzy graph G = P₆ such that by eliminating the particular fuzzy edge which splits into two fuzzy graphs G′ and G″. If the number of vertices and edges present in two fuzzy graphs are equal then MWI (G′) = MWI (G″) < MWI (G).

Theorem 10. Let G be the connected fuzzy graphs such that the elimination of fuzzy edges forms disconnected. Then

$\begin{matrix} MWI (G^{'}) + MWI (G^{″}) + \dots + MWI (G^{n}) \\ < MWI (G) \end{matrix}$ (7)

Proof. Since G - uv is disconnected and by the Theorem 8 for two disconnected graphs satisfies MWI (G′) < MWI (G) and MWI (G″) < MWI (G) with m′ ≤ m and m″ ≤ m. Suppose more number of disjoint graphs are obtained by eliminating the set of edges in fuzzy graphs produces disconnected graphs G′, G″, G^″′, … Gⁿ. Then MWI (G′) < MWI (G″) < MWI (G^″′) … < MWI (Gⁿ) where m′ ≤ m″ ≤ m^″′ … ≤ mⁿ. Adding the Modified Wiener indices of G, G … Gⁿ we get the Equation (7) as $MWI (G^{'}) + MWI (G^{″}) + \dots + MWI (G^{n}) < MWI (G) .$

Hence the theorem is proved.

Example 11. Consider the fuzzy graph G as shown in the Fig. 4 and 5, by removing the fuzzy edge then the two fuzzy graphs having MWI (G′) < MWI (G″). Then MWI (G′) + MWI (G″) < MWI (G). That is MWI (G) =5.298, when the fuzzy edge 0.4 is removed and split into two fuzzy graphs as G and G. Then MWI (G) =0.121 and MWI (G″) =1.7565. MWI (G′) + MWI (G″) =0.121 + 1.7565 < 5.298 = MWI (G) .

Fig. 4

Fuzzy graph G.

Fig. 5

Fuzzy graph G -{ 0.4 }.

4.3 Hyper wiener index in fuzzy graph

The Hyper Wiener Index is also a topological index which is used in the field of biochemistry. The Hyper Wiener index was introduced by M. Randić. Now, we define the Hyper Wiener index in terms of fuzzy graph as follows.

Definition 12. The Hyper Wiener index WWI (G) of a fuzzy graph G is defined as $\begin{matrix} WWI (G) = \frac{1}{4} \sum_{u, v \in σ} (σ (u) σ (v) d_{s} (u, v) \\ + (σ (u) σ (v) d_{s} (u, v))^{2}) \end{matrix}$

Where d_s (u, v) is the minimum sum of weights of distance from u to v.

Example 13. Consider the fuzzy graph G = (V, σ, μ) with σ = { a, b, c, d, e } = 1, μ (a, b) = μ (b, d) = μ (c, e) = 0.7, μ (c, d) = μ (b, e) = 0.5, μ (a, c) = 0.8 and μ (a, c) = 0.6. There exists unique distance between every pair of vertices in G. Then by calculation, WWI (G) = 7.87 .

4.3.1 Boundedness for hyper wiener index

Here, discuss about the lower bound for the Hyper Wiener index.

Theorem 14. Let G be a fuzzy graph with order n and size m. If p is the least membership value present in the fuzzy graph G, then $WWI (G) < mnp$ (8)

Proof. Let G be a fuzzy graph with order n and size m where the membership value of each vertex and edge lies in the interval [0, 1] and (0, 1) respectively. Since,p is the least membership value present in the fuzzy graph G, such that u, v ∈ σ, then d_s (u, v) ≥ p, d_s (u, v) ≤ m and d_s (u, v) ≤ n. Thus, ∑d_s (u, v) ≤ mnp. Hence, WWI (G) < mnp. Hence the thoerem is proved.

Special case 15. Let G be a fuzzy graph with order n and size m repsectively. If σ (u) = μ (u, v) = 1, forallu, v ∈ G . Then, WWI (G) ≤ 3n (n - 1) - 4m ≤ mn.

This case is true only for the fuzzy graphs whose membership value of vertices and edges are to be 1.

4.4 Schultz and Gutman indices in fuzzy graph

Now, another set of important indices in topology are Schultz and Gutman indices. Schultz index was introduced by H.P. Schultz through the paper [7] in 1989. The Schultz index and Gutman index are defined as follows:

Definition 16. The Schultz index SCI (G) of the fuzzy graph is $SCI (G) = \sum_{u, v \in σ} σ (u) σ (v) d (u, v) (d (u) + d (v))$ where d (u) andd (v) denotes the degree of u and v respectively.

Importance of Schultz is derived by adding degrees of the vertices of the fuzzy graph. By the same assumption Gutman introduced another topological index and which is discussed in fuzzy graph explained below.

Definition 17. The Gutman index GI (G) of the fuzzy graph is $GI (G) = \sum_{u, v \in σ} σ (u) σ (v) d (u, v) d (u) d (v)$ for every edge in G.

Example 18. Consider the fuzzy graph G = (V, σ, μ) with σ = {a, b, c, d, e} =1, μ (a, b) =0.6, μ (b, d) = μ (d, e) =0.5, μ (a, c) =0.75, μ (b, c) =0.3, and μ (c, e) =0.4 shown in Fig. 7. There exists unique distance between every pair of vertices in G.

Fig. 6

Example for Hyper Wiener index.

Fig. 7

Fuzzy graph.

Then the calculated for Schultz value is 15.1225 and Gutman index is 24.857.

4.4.1 Boundedness for Schultz indices

In this section, the lower bound for the Schultz indices are discussed as follows:

Theorem 19. Let G be a fuzzy graph with m and n are order and size respectively. Then $SCI (G) \geq (δ (G)) mnp, if | m^{*} | < | n^{*} |$ (7) $SCI (G) \geq \frac{1}{2} (Δ (G) mnp), if | m^{*} | \geq | n^{*} |$ (8)

Where |m^*|and|n^*| represents the number of strong edges and weak edges in G, p be the least membership value of the edge present in G and δ (G), Δ (G) represent the minimum and maximum degree value of the fuzzy graph G.

4.5 Fuzzy planar index

Definition 20. The Planarity index PI (G) of the fuzzy graph is defined as the given fuzzy graph has to reduce the intersection of fuzzy edges in it. If the intersection of fuzzy edges is in G then deleting any one of the fuzzy edge and it can be calculated as follows $PI (G) = min {\begin{matrix} \frac{1}{2} \sum_{G - uv} σ (u) σ (v) d_{s} (u, v), \\ \frac{1}{2} \sum_{G - xy} σ (u) σ (v) d_{s} (u, v) \end{matrix}}$

Example 21. Consider the fuzzy graph G = (V, σ, μ) of K₅ with σ = {a, b, c, d, e} =1, μ (a, b) =0.7, μ (a, c) = μ (c, e) =0.8, μ (a, d) = μ (b, d) =0.2, μ (d, c) =0.9, μ (b, e) =0.5, μ (a, c) =0.8 and μ (b, c) =0.3. Thus, for the above example, PI (G) =2.51.

Theorem 22. Every planar index value is the same for fuzzy planar graphs and its dual.

Proof. Let G be the fuzzy planar graph with the planarity value as 1. Then there exists a dual fuzzy graph such that crossing is not allowed. There is a one-to-one correspondence between the two graphs whose planar index value are the same.

5 Topological indices in the fuzzy graph – Part 2

One of the major concepts in topological indices is the Zagreb index. It is valuable to know the parameter’s complexity in a chemical system, biological system, etc. Now, this special index brought to the fuzzy graph theory such that its objectives and properties are observed from the following definitions.

5.1 Zagreb indices in fuzzy graphs

Definition 23. Let G be the fuzzy graph with non-empty vertex set. The first Zagreb index is denoted by $M (G)$ and defined as $M (G) = \sum_{u_{i} \in σ} (u_{i}) d^{2} (u_{i}), \forall u_{i} \in V$ where d (u_i) is the degree of the vertices.

Definition 24. The second kind of Zagreb index is denoted by $M^{*} (G)$ and defined as $M^{*} (G) = \frac{1}{2} \sum_{(u_{i}, u_{j}) \in μ} σ (u_{i}) d (u_{i}) σ (u_{j}) d (u_{j}),$ where i ≠ j.

Example 25. Let G be the fuzzy graph as shown in the Fig. 9 such that V ={ a, b, c, d } where σ (a) = 0.5, σ (b) = 0.3, σ (c) = 0.8, σ (d) = 0.75. The edge set contains μ (a, b) = 0.25, μ (b, d) = 0.1, μ (a, c) = 0.4 and μ (b, c) = 0.2. The Zagreb’s indices of G on first and second kind are $M (G) = \sum_{u_{i} \in σ} (u_{i}) d^{2} (u_{i}), \forall u_{i} \in V,$ where i = {1, 2, 3, 4} $\begin{matrix} M (G) = (0.5) {(0.65)}^{2} + (0.3) {(0.55)}^{2} \\ + (0.8) {(0.6)}^{2} + (0.75) {(0.1)}^{2} = 0.5975 \\ M^{*} (G) = \frac{1}{2} \sum_{(u_{i}, u_{j}) \in μ} σ (u_{i}) d (u_{i}) σ (u_{j}) d (u_{j}) \end{matrix}$ Where i ≠ j and (u_i, u_j) ∈ μ with v_i, v_j ∈ V, i, j ={ 1, …, 4 } $\begin{matrix} M^{*} (G) = 0.5 {(0.5 \times 0.65 \times 0.3 \times 0.55) \\ + (0.3 \times 0.55 \times 0.8 \times 0.6) + (0.6 \times 0.8 \times 0.5 \\ \times 0.65) + (0.3 \times 0.55 \times 0.75 \times 0.1)} = 0.1506 \end{matrix}$

Fig. 8

Example for Planar index.

Fig. 9

Example for Zagreb indices.

Corollary 26. Let G be the fuzzy graph with u ∈ G and H be the subgraph of G by H = G -{ u }. Then $M (G - {u}) < M (G)$ and $M^{*} (G - {u}) < M^{*} (G)$ .

Corollary 27. Every fuzzy graph whose Zagreb indices satisfies $M (G) \geq M^{*} (G)$ .

5.1.1 Boundedness for zagreb indices

Theorem 28. Let G be the fuzzy graph and bounds for Zagreb indices are given by

$M (G) < \frac{m^{2} + n^{2}}{2}$

$mn < M^{*} (G) < m + n$

Proof. Case(i). Let G be the fuzzy graph with m and n denotes the order and size of the fuzzy graphs respectively. Suppose that the fuzzy graph is connected by the finite number of line segments. Each line segment has two fuzzy vertices and one adjacent edge. The degree of each vertices in G is denoted u_iwhere i = 1, 2, …, n. Therefore, each degree of the vertices has d (u_i) < m, ∀ u_i ∈ V. Then squaring on both sides, d² (u_i) < m². Then $\sum_{i = 1}^{n} σ (u_{i}) d^{2} (u_{i}) < m^{2}$ (8) and therefore $\sum_{i = 1}^{n} σ (u_{i}) d^{2} (u_{i}) < n^{2}$ (9)

Hence adding Equations (8) & (9), we get $\sum_{i = 1}^{n} σ (u_{i}) d^{2} (u_{i}) < \frac{m^{2} + n^{2}}{2}$ (10)

Case(ii). The Zagreb index of second kind’s boundedness has the fuzzy graph G considered to be finite number of line segments. Therefore $M^{*} (G) < 2 m$ and $M^{*} (G) < 2 n$ which implies $M^{*} (G) < m + n$ . We know thats d (u_i) < m and d (u_i) < n. Hence, there exist mn < σ (u_i) d (u_i) σ (u_j) d (u_j), for every edge in G which implies that $mn < \sum_{i = 1}^{n} σ (u_{i}) d (u_{i}) σ (u_{j}) d (u_{j})$ and which is the lower bound for the Zagreb second kind.

5.2 Additional indices in fuzzy graphs

Definition 29. The Randić index in fuzzy graph G, $(R (G))$ is defined as $R (G) = \frac{1}{2} \sum_{(u_{i}, u_{j}) \in μ} (σ (u_{i}) σ (v_{j}) d (u_{i}) d (v_{j}))^{- 1 / 2},$ where ∀i ≠ j.

Definition 30. The Harmonic index of G, $(H (G))$ is defined as $H (G) = \frac{1}{2} \sum_{(u_{i}, u_{j}) \in μ} (σ (u_{i}) d (u_{i}) + σ (u_{j}) d (u_{j}))^{- 1}$ where ∀i ≠ j.

5.3 Some important properties on topological indices in fuzzy graphs

Here we discuss the properties of the topological indices.

Theorem 31. Let G and H be the two fuzzy graphs. If the two fuzzy graphs are isomorphic to each other then the topological indices values of two fuzzy graphs are equal.

Proof. Let G = (V_G, σ_G, μ_G) and H = (V_H, σ_H,μ_H) be isomorphic fuzzy graphs. Hence there exists an identity function I_σ : σ_G (u) → σ_H (u^*), for every u ∈ V_G there exist u^* ∈ V_H as well as $I_μ : μ_G (u, v) → μ_H (u^*, v^*) (i.e.,) each membership value of vertices and edges of G coincides with the fuzzy graph H. Therefore, the graph structure may differ but collection of vertices and edges are same gives the equal topological indices value.

Corollary 32. If the topological values of G and H are same, then the two fuzzy graphs need not to be isomorphic.

Theorem 33. Let G be the connected fuzzy graph and μ (u, v) be the membership value of the chosen edge of G such that removal of certain edges from G splits into two fuzzy graphs such that whose vertex set satisfies |V_G′| < |V_G″| and therefore its disconnected. Then

$M (G^{'}) < M (G^{″})$

$M^{*} (G^{'}) < M^{*} (G^{″})$

$R (G^{'}) > R (G^{″})$

$H (G^{'}) > H (G^{″})$

where |V_G′|and|V_G″| denotes the cardinality of G′ and G″ respectively.

Proof. Let G be the connected fuzzy graph where splitted into two fuzzy graphs G′ and G″ by removing the chosen membership value of the edge in G. By Corollary 1, $M (G - {u}) < M (G),$ and therefore, we get, $M (G^{'}) < M (G^{″}) < M (G)$ which implies that $M (G^{'}) < M (G^{″})$ . Hence, the theorem is proved.

The remaining cases are trivially true by the same process.

Corollary 34. Let us consider a fuzzy graph of G = P₆ in Fig. 10 elimination of the particular fuzzy edge which split it into two fuzzy graphs G′ and G″. The two fuzzy graphs are isomorphic to each other and the distance matrix of two fuzzy graphs are same. Then

$M (G^{'}) = M (G^{″})$

$M^{*} (G^{'}) = M^{*} (G^{″})$

$R (G^{'}) = R (G^{″})$

$H (G^{'}) = H (G^{″})$

Fig. 10

Example for Corollary 35.

Proof. Let G be the fuzzy graph and deletion of certain edges gives the fuzzy graph into as disconnected fuzzy graphs. Suppose there is an isomorphic on vertices and edges of two fuzzy graphs but the structurally not equal then the two fuzzy graphs have different distance matrix such that the topological values of two fuzzy graphs are not equal which is a contraction to the assumption and its necessary that the two fuzzy graphs same distance matrix.

Theorem 35. Let G be the connected fuzzy graphs such that the elimination of fuzzy edges forms disconnected. Then,

$M (G^{'}) + M (G^{″}) + \dots + M (G^{n}) \leq M (G)$

$M^{*} (G^{'}) + M^{*} (G^{″}) + \dots + M^{*} (G^{n}) \leq M^{*} (G)$

$R (G^{'}) + R (G^{″}) + \dots + R (G^{n}) \geq R (G)$

$H (G^{'}) + H (G^{″}) + \dots + H (G^{n}) \geq H (G)$

EI (G′) + EI (G″) + … + EI (Gⁿ) ≤ EI (G)

Proof. By the Corollary 26 and Theorem 34, it’s trivially true for each case.

5.4 Relationship between operations on indices of fuzzy graphs

Here, we discuss about the union and intersection of the fuzzy graphs and the results on topological indices.

Corollary 36. Let G and H be the two fuzzy graphs. The union of two fuzzy graphs (i.e.,) G ∪ H is

$M (G \cup H) \geq \max {M (G), M (H)}$

$M^{*} (G \cup H) \geq \max {M^{*} (G), M^{*} (H)}$

$R (G \cup H) \leq \max {R (G), R (H)}$

$H (G \cup H) \leq \max {H (G), H (H)}$

Corollary 37. Let G and H be the two fuzzy graphs. The intersection of two fuzzy graphs (i.e.,) G ∩ H is defined on topological indices are as follows:

$M (G \cap H) \leq \min {M (G), M (H)}$

$M^{*} (G \cap H) \leq \min {M^{*} (G), M^{*} (H)}$

$R (G \cap H) \geq \min {R (G), R (H)}$

$H (G \cap H) \geq \min {H (G), H (H)}$

Theorem 38. Let G_i ∀i = 1, 2, …, n be the fuzzy graphs. The union of fuzzy graphs is denoted by $\cup_{i = 1}^{n} G_{i}$ . The relationship between the union and the sum of the topological indices in fuzzy graphs satisfies the following:

If $M (\cup_{i = 1}^{n} G_{i}) \geq \max {M (G_{i})$ , ∀i = 1, 2, …n}, then $M (\cup_{i = 1}^{n} G_{i}) \leq \sum_{i = 1}^{n} M (G_{i})$ , ∀i = 1, 2, … n.

If $M^{*} (\cup_{i = 1}^{n} G_{i}) \geq \max {M^{*} (G_{i}), \forall i = 1, 2, \dots n}$ , then $M^{*} (\cup_{i = 1}^{n} G_{i}) \leq \sum_{i = 1}^{n} M^{*}$ (G_i) , ∀ i = 1, 2, … n.

If $R (\cup_{i = 1}^{n} G_{i}) \leq \max {R (G_{i})$ , ∀i = 1, 2, …n}, then $R (\cup_{i = 1}^{n} G_{i}) \leq \sum_{i = 1}^{n} R (G_{i}), \forall i = 1, 2, \dots n$ .

If $H (\cup_{i = 1}^{n} G_{i}) \leq \max {H (G_{i})$ , ∀i = 1, 2, …n}, then $H (\cup_{i = 1}^{n} G_{i}) \leq \sum_{i = 1}^{n} H (G_{i}), \forall i = 1, 2, \dots n$ .

Proof. Let G_i = (V_{G
_i}, σ_{G
_i}, μ_{G
_i}) be the fuzzy graph and i = 1, 2, …, n. We know that $M (\cup_{i = 1}^{n} G_{i}) \geq \max {G_{i}, i = 1, 2, \dots, n}$ by the Corollary 37 and its obviously true.

Case(i): Let G_i = (V_{G
_i}, σ_{G
_i}, μ_{G
_i}) and G_{i
^*} = (V_{G
_{i
^*}}, σ_{G
_{i
^*}}, μ_{G
_{i
^*}}) be the two fuzzy graphs. Consider V_{G
_i} ≠ V_{G
_{i
^*}} ≠ φ, then V_{G
_i} ∩ V_{G
_{i
^*}} = φ which implies that σ_{G
_i} ∩ σ_{G
_{i
^*}} = φ and μ_{G
_i} ∩ μ_{G
_{i
^*}} = φ. Then the Zagreb index of first kind defined on union operation is given by $M (G_{i} \cup G_{i^{*}}) = M (G_{i}) + M (G_{i^{*}})$ . Hence, $M (\cup_{i = 1}^{n} G_{i}) = \sum_{i = 1}^{n} M (G_{i}), \forall i = 1, 2, \dots n$ , which implies that fuzzy graphs unique.

Case(ii): Suppose G_i = (V_{G
_i}, σ_{G
_i}, μ_{G
_i}) and G_{i
^*} = (V_{G
_{i
^*}}, σ_{G
_{i
^*}}, μ_{G
_{i
^*}}) be the two fuzzy graphs. Consider some of the vertices in V_{G
_i}andV_{G
_{i
^*}} are common, (i.e.,) σ_{G
_i} ∩ σ_{G
_{i
^*}} ≠ φ but there exists as μ_{G
_i} ∩ μ_{G
_{i
^*}} = φ. Then the two fuzzy graphs G_i and G_{i
^*} are not equal and satisfies the condition as $M (\cup_{i = 1}^{n} G_{i}) = \sum_{i = 1}^{n} M (G_{i}), \forall i = 1, 2, \dots n$ , which implies that fuzzy graphs $G_{i}^{'} s$ are unique.

Case(iii): Let G_i = (V_{G
_i}, σ_{G
_i}, μ_{G
_i}) and G_{i
^*} = (V_{G
_{i
^*}}, σ_{G
_{i
^*}}, μ_{G
_{i
^*}}) be the two fuzzy graphs. There exist σ_{G
_i} (x_j) = σ_{G
_{i
^*}} (y_j) and σ_{G
_i} (x_{j
^*}) = σ_{G
_{i
^*}} (y_{j
^*}) with μ_{G
_i} (x_j, x_{j
^*}) = μ_{G
_{i
^*}} (y_j, y_{j
^*}). Suppose, adjacent to the vertices of G_i and G_{i
^*} are different. Then the two fuzzy graphs have different $M (G_{i})$ and $M (G_{i^{*}})$ values which shows that $M (\cup_{i = 1}^{n} G_{i}) \leq \sum_{i = 1}^{n} M (G_{i}), \forall i = 1, 2, \dots n$ .

Case(iv): Let G_i = (V_{G
_i}, σ_{G
_i}, μ_{G
_i}) and G_{i
^*} = (V_{G
_{i
^*}}, σ_{G
_{i
^*}}, μ_{G
_{i
^*}}) be the two fuzzy graphs. Consider one of the fuzzy graph structures coincides with other but not equal (i.e.,) G_{i
^*} - { u } = G_i where u ∈ V_{G
_{i
^*}}. Then $M (\cup_{i = 1}^{n} G_{i}) = \max {M (G_{i}), M (G_{i^{*}}) + {u})$ , ∀i = 1, 2, … n}. The sum operation is as follows: $M (\cup_{i = 1}^{n} G_{i}) < \sum_{i = 1}^{n} M (G_{i}), \forall i = 1, 2, \dots n .$

Hence, the theorem is proved for Zagreb index on first kind.

By following the same procedure, the relation between the union and sum of the topological indices such as Zagreb second kind, Randić and Harmonic indices are proved.

Theorem 39. Let G_i ∀i = 1, 2, …, n be the fuzzy graphs. The intersection of fuzzy graphs is denoted by $\cap_{i = 1}^{n} G_{i}$ . The relationship between the union and the sum of the topological indices in fuzzy graphs satisfies the following:

If $M (\cap_{i = 1}^{n} G_{i}) \leq \max {M (G_{i})$ , ∀i = 1, 2, …n}, then $M (\cap_{i = 1}^{n} G_{i}) \leq \sum_{i = 1}^{n} M (G_{i}),$ ∀i = 1, 2, … n.

If $M^{*} (\cap_{i = 1}^{n} G_{i}) \leq \max {M^{*} (G_{i}), \forall i = 1, 2, \dots n}$ , then $M^{*} (\cap_{i = 1}^{n} G_{i}) \leq \sum_{i = 1}^{n} M^{*}$ (G_i) , ∀ i = 1, 2, … n.

If $R (\cap_{i = 1}^{n} G_{i}) \geq \max {R (G_{i})$ , ∀i = 1, 2, …n}, then $R (\cap_{i = 1}^{n} G_{i}) \geq \sum_{i = 1}^{n} R (G_{i}), \forall i = 1, 2, \dots n$ .

If $H (\cap_{i = 1}^{n} G_{i}) \geq \max {H (G_{i})$ , ∀i = 1, 2, …n}, then $H (\cap_{i = 1}^{n} G_{i}) \geq \sum_{i = 1}^{n} H (G_{i}), \forall i = 1, 2, \dots n$ .

Proof. Let G_i = (V_{G
_i}, σ_{G
_i}, μ_{G
_i}) be the fuzzy graph and i = 1, 2, …, n. We know that $M (\cap_{i = 1}^{n} G_{i}) \leq \min {G_{i}, i = 1, 2, \dots, n}$ by the following the Corollary 37 and its obviously true.

Case(i): Let G_i = (V_{G
_i}, σ_{G
_i}, μ_{G
_i}) and G_{i
^*} = (V_{G
_{i
^*}}, σ_{G
_{i
^*}}, μ_{G
_{i
^*}}) be the two fuzzy graphs. Consider V_{G
_i} ≠ V_{G
_{i
^*}} ≠ φ, then V_{G
_i} ∩ V_{G
_{i
^*}} = φ which implies that σ_{G
_i} ∩ σ_{G
_{i
^*}} = φ and μ_{G
_i} ∩ μ_{G
_{i
^*}} = φ. Then the Zagreb index of first kind defined on intersection is given by $0 = M (G_{i} \cap G_{i^{*}}) < M (G_{i}) + M (G_{i^{*}})$ . Hence, $M (\cap_{i = 1}^{n} G_{i}) < \sum_{i = 1}^{n} M (G_{i}), \forall i = 1, 2, \dots n$ , which implies that fuzzy graphs $G_{i}^{'} s$ are unique.

Case(ii): Let G_i = (V_{G
_i}, σ_{G
_i}, μ_{G
_i}) and G_{i
^*} = (V_{G
_{i
^*}}, σ_{G
_{i
^*}}, μ_{G
_{i
^*}}) be the two fuzzy graphs. Consider one of the fuzzy graph structures coincides with other but not equal (i.e.,) G_{i
^*} - { u } = G_i where u ∈ V_{G
_{i
^*}}. Then $M (\cap_{i = 1}^{n} G_{i}) = \min {M (G_{i}), M (G_{i^{*}}) + {u})$ , ∀i = 1, 2, … n}. The sum operation is as follows: $M (\cup_{i = 1}^{n} G_{i}) < \sum_{i = 1}^{n} M (G_{i}), \forall i = 1, 2, \dots n .$

Hence, the theorem is proved for Zagreb index on first kind.

By following the same procedure, the relation between the union and sum of the topological indices such as Zagreb second kind, Randić and Harmonic indices are proved.

5.5 Algorithms for topological indices

This section provides to understand the topological indices in the algorithm method.

5.5.1 Algorithm for Distance matrix

Let G be the fuzzy graph with σ vertices and μ edges.

Input: The vertex set, edge set and adjacency list for the given fuzzy graph G

Output: M_dst = DistanceMatrix = A_ij

Step 1: Calculate the distance for every pair of vertices between them.

Step 2: If there is no μ-value between any two vertices, then A_ij = 0.

Step 3: If i ≠ j, then A_ij ≠ 0 or otherwise A_ii = 0.

Step 4: Calculate the shortest distance between the pair of vertices by passing through the lesser membership values. That is $A_{ij} = min {A_{ij}, A_{ic} + A_{ck}, \dots} .$

5.5.2 Algorithm for MWI

Let G be the fuzzy graph with membership value of σ vertices and μ edges.

Input: Distance Matrix

Output: MWI (G) = ModifiedWienerindex

Step 1: σ (u) σ (v) d (u, v) is the product of each distance value with the adjacent vertices’ membership value.

Step 2: Adding distance value for every edge present in the fuzzy graph G.

Step 3: Finally, Divides by 2 gives the MWI (G)

By following the same procedures as MWI (G), we get the remaining topological indices in fuzzy graphs.

6 Application

J.N. Moderson et al. [18] presented applications in the area of human trafficking, internet routing, etc. The topological indices in fuzzy graphs enable accuracy in the calculation because the fuzzy edges are depended only on the membership value of adjacent vertices. Moreover, this type is applicable in the Distribution Network Representation, where informatics systems for supervising and monitoring it.

Likewise, in cheminformatics topological indices are familiar notion can be used as quantitative structure-activity and the structure-property relationship are helps in the study of nanomaterial and for biological activity.

Various problems in real-time can be converted into a fuzzy graph rather than crisp so that exactness is preserved. The topological indices made useful in the field of computer science, networking, scheduling, etc. because it concerns the in the distance and strength of the vertices and edges shows the exactness.

6 Conclusion

Topological indices are familiar in Graph theory, mainly it has usefulness in the field of molecular science, also it’s applied in the field of computer science, networking, scheduling, etc. In fuzzy mathematics, this concept is new and can also be applied in the field of chemical sciences. In the normal graph, the presence of vertices and edges has weightage value as 1 but in the fuzzy graph, the membership value of vertices and edges has in the range of [0, 1]. Our aim in this paper is to show that the topological indices are defined and studied in terms of the fuzzy graphs. In this paper, definitions for indices such as Modified Wiener index, Hyper-Wiener index, Schultz index, Gutman index, Zagreb indices, Harmonic index, and Randić index are shown with illustrations. Here Planarity index value is new to the topological indices’ family is discussed. The deletion of fuzzy edges produces a relation in topological indices are studied for MWI. Disconnected graphs are obtained by deletion of the fuzzy edges are shown with example and results on this notion are presented. Finally, set operations such as union and intersection are shown for the Zagreb indices, Randić, and Harmonic indices with some of the results are discussed.

In future, the different topological indices exist can be defined and shown in the fuzzy graph and application to chemical structures can be extended in the upcoming papers.

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