First Zagreb index on a fuzzy graph and its application

Abstract

The Zagreb index (ZI) is a very important graph parameter and it is extensively used in molecular chemistry, spectral graph theory, network theory and several fields of mathematics and chemistry. In this article, the first ZI is studied for several fuzzy graphs like path, cycle, star, fuzzy subgraph, etc. and presented an ample number of results. Also, it is established that the complete fuzzy graph has maximal first ZI among n-vertex fuzzy graphs. Some bounds of first ZI are discussed for Cartesian product, composition, union and join of two fuzzy graphs. An algorithm has been designed to calculate the first ZI of a fuzzy graph. At the end of the article, a multi-criteria decision making (MCDM) method is provided using the first ZI of a fuzzy graph to find the best employee in a company. Also a comparison is provided among related indices on the result of application and shown that our method gives better results.

Keywords

Fuzzy graph First Zagreb index MCDM

Abbreviations

Fuzzy set

FSS

Fuzzy subset

Fuzzy graph

Crisp graph

FGT

Fuzzy graph theory

FSG

Fuzzy subgraph

PFSG

Partial fuzzy subgraph

CONFG

Connected fuzzy graph

CFG

Complete fuzzy graph

CPFG

Completion fuzzy graph

Topological index

Zagreb index

Molecular Graph

Membership value

CPG

Cartesian product graph

Union graph

1 Introduction

1.1 Research background

Nowadays, due to various applications of fuzzy graph theory (FGT), a huge number of researchers are working on topological indices (TIs). In 1965, the idea of fuzzy set (FS) was first introduced by Zadeh [52]. Motivated by this, in 1975, Rosenfeld [36] defined the fuzzy graph (FG). At the same time, Yeh et al. [51] also introduced the FG independently and defined some connectivity parameters of a FG and their applications. In [48, 49], Sunitha et al. studied the fuzzy block (FB), fuzzy bridge (FBg), fuzzy cut vertex, fuzzy tree (FT), fuzzy forest, partial fuzzy subgraph (PFSG), fuzzy subgraph (FSG), complete fuzzy graph (CFG), etc. Samanta and Pal discussed a fuzzy planar graph in [41]. In [37 –39] Sahoo and Pal provided a product of intuitionistic fuzzy graphs. In [31, 32] interval-valued FG is studied by Rashmanlou and Pal. Degree of a vertex in a FG is also discussed in [29]. In [1–6 , 43] one can see more details on FGT. In the field of molecular chemistry, TIs are molecular descriptors which are calculated on the molecular graph (MG) of a chemical compound. These TIs are numerical quantities of a graph which describes its topology. In [26 –28] Mondal et al. studied some neighbourhood degree-based TIs. Zagreb index (ZI) is one such TIs which is degree-based TI and introduced by Gutman and Trinajstic in 1972 [12] and this TI is used to calculate π-electron energy of a conjugate system. In [44] Sarkar et al. studied ZI of some graph operations and also they calculated general ZIs of some carbon structure in [45]. In [7 , 47], one can see for further work related to this topic. In [11 , 50], one can see for different types of TIs. But those indices are defined in a crisp graph. As fuzzy graphs is the generalization of crisp graphs and many real life problems cannot be handled by crisp graphs, those indices in fuzzy graphs have more applications. In this article, the first Zagreb index for fuzzy graphs is introduced which is a more generalization of the first Zagreb index for crisp graphs and it is extensively used in spectral fuzzy graph theory, fuzzy network theory, fuzzy graph theory and several fields of fuzzy mathematics and chemistry. This index is studied for both theoretical and application point of view also.

1.2 Research question

In this paper, the following questions are discussed: (i) In [13] Hosamani and Basavanagoud provided the upper bound of the first Zagreb index for crisp graphs. What is the upper bound of the first Zagreb index for fuzzy graphs?

(ii)What are the exact values/bounds of the first Zagreb index for path, cycle, star, complete fuzzy graph, etc.?

(iii) For n-vertex crisp graph, complete graph has maximum first Zagreb index. Is this result true for fuzzy graphs also?

(iv) In [21] Khalifeh et al. has given the exact expressions of the first Zagreb index for crisp graph operations (Cartesian product, composition, join, union). What are the exact expressions of the first Zagreb index for fuzzy graph operations (Cartesian product, composition, join, union)?

(v) What is the application of this index? Is this index providing better results for this application among other indices?

1.3 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 modeled using crisp graphs. In these cases, to handle such a situation, those topological indices are needed to define a fuzzy graph. In this paper, modification of the first Zagreb index is defined and given some bounds for various fuzzy graphs like path, cycle, star, complete fuzzy graph (CFG), etc. In this paper, n-vertex maximal fuzzy graph is determined with respect to the first Zagreb index. Some bounds of first ZI are discussed for Cartesian product, composition, union and join of two fuzzy graphs. An algorithm has been also designed to calculate the first ZI of a fuzzy graph. At the end of the article, a MCDM method is provided using the first Zagreb index of a fuzzy graph to find the best employee in a company. Also a comparison is provided among related indices on the result of application and shown that our method gives better results.

1.4 Structure of the study

Structure of the article is as follows: Section 2 provides basic definitions which are essential to develop our main results. In section 3, first ZI of a FG is studied and provides the value of it for different FGs and also discusses their bounds. In Section 4, some bounds of first ZIs are discussed during FG operations (Cartesian product, composition, union, join). In Section 5, an algorithm is designed to compute the first ZI of a FG. An application of ZI in MCDM problem has been considered in Section 6. In Section 7, a comparison is provided among related indices on the result of application and shown that our method gives better results.

2 Preliminaries

In this Section, we give some basic definitions which are essential to develop our results. Let X be a universal set. A FS S on X is a mapping Ω : X → [0, 1]. Here Ω is called the membership function of the FS S. Generally a FS is denoted by S = (x, Ω). Let X (≠ φ) be a given finite set. The FG is a triplet, $G = (V, Ψ, Ω)$ , where $V$ is nonempty finite subset of X with $Ψ : V \to [0, 1]$ and $Ω : V \times V \to [0, 1]$ satisfying Ω (x, y) ≤ Ψ (x) ∧ Ψ (y), where ∧ represents the minimum. The set $V$ is the set of vertices and $E : = {(x, y) : Ω (x, y) > 0}$ is the set of edges of the FG. Ψ (x) represents the MV of the vertex x and Ω (x, y) represents the membership value (MV) of the edge (x, y) (or simply xy). Let $G = (V, Ψ, Ω)$ be a FG. Then $H = (V^{'}, Ψ^{'}, Ω^{'})$ is called partial fuzzy subgraph (PFSG) of the FG $G$ if $V^{'} \subset V, Ψ^{'} (x) \leq Ψ (x), Ω^{'} (x, y) \leq Ω (x, y)$ for all $x, y \in V^{'}$ . If Ψ′ (x) = Ψ (x) and Ω′ (x, y) ≤ Ω (x, y) for all $x, y \in V^{'}$ then $H$ is called fuzzy subgraph (FSG) of the FG $G$ . For $x \in V$ , we denote $G_{x}$ is a FSG of the FG $G$ with Ψ (x) =0 and for $xy \in E$ , $G_{xy}$ represents the FSG of the FG $G$ with Ω (xy) =0. Let x₀, x₁, ⋯ , x_n be distinct vertices of a FG $G$ . Then the sequence of vertices P (x₀x₁ ⋯ x_n) is called a path in $G$ if Ω (x_i, x_i+1) ≠0 for i = 0, 1, ⋯ , n - 1. For this case the length of the path is n. The path, P is called a cycle if Ω (x₀, x_n) >0. Let $G$ be a FG. $G$ is called a connected FG if for any two vertices v_i and v_j, there exists a path containing v_i and v_j. Let x₀, x₁, ⋯ , x_n be the vertices of a FG $G$ . Then $G = (x_{0}, x_{1}, x_{1} \dots x_{n})$ is called a star if Ω (x₀, x_i) ≠0 for i = 0, 1, ⋯ , n. Here x₀ is called the center of the star. Let $G = (V, Ψ, Ω)$ be a FG. $G$ is called a complete fuzzy graph (CFG) if for all $x, y \in V, Ω (x, y) = Ψ (x) \land Ψ (y)$ . Two FGs $G_{1} = (V_{1}, Ψ_{1}, Ω_{1})$ and $G_{2} = (V_{2}, Ψ_{2}, Ω_{2})$ are called isomorphic if there exist a bijective map $h : V_{1} \to V_{2}$ with any $x, y \in V_{1}, Ψ_{1} (x) = Ψ_{2} (h (x))$ and Ω₂ (h (x) , h (y)) = Ω₁ (x, y). Let $v \in V$ be a vertex of a FG $G$ . Then the degree of v is denoted by $d_{G} (v)$ or simply d (v) and defined as $d (v) = \sum_{x \in V} Ω (xv) .$ Let $Δ (G)$ or Δ be the maximum degree of $G$ and defined as $Δ = \lor_{v \in V} d (v)$ and $δ (G)$ or δ be the minimum degree of G and defined as $δ = \land_{v \in V} d (v)$ . The total degree of $G$ is denoted by $T (G)$ or simply T, i.e. $T = \sum_{v \in V} d (v) = 2 \sum_{uv \in E} Ω (uv) \leq 2 \times no . of edges$ . Throughout this article, we consider, $G_{1} = (V_{1}, Ψ_{1}, Ω_{1})$ with n₁-vertices, m₁-edges and $G_{2} = (V_{2}, Ψ_{2}, Ω_{2})$ with n₂-vertices, m₂-edges be two FGs and $Δ_{1} = Δ (G_{1}), Δ_{2} = Δ (G_{2}), δ_{1} = δ (G_{1}), δ_{2} = δ (G_{2}) .$ Cartesian product of two FGs are defined below:

Definition 2.1. [29] Cartesian product of $G_{1}$ and $G_{2}$ is a FG $G_{1} \times G_{2} = (V, Ψ, Ω)$ , where $V = V_{1} \times V_{2}$ , $(u, v), (u_{1}, v_{1}), (u_{2}, v_{2}) \in V, Ψ (u, v) = \land {Ψ_{1} (u), Ψ_{2} (v)}$ and $Ω ((u_{1}, v_{1}), (u_{2}, v_{2})) = {\begin{matrix} \land {Ψ_{1} (u_{1}), Ω_{2} (v_{1}, v_{2})} \\ if u_{1} = u_{2}, v_{1} v_{2} \in E_{2} \\ \land {Ψ_{2} (v_{1}), Ω_{1} (u_{1}, u_{2})} \\ if u_{1} u_{2} \in E_{1}, v_{1} = v_{2} \\ 0 otherwise . \end{matrix}$ Composition of two FGs are defined below:

Definition 2.2. [29] Let $G_{1} = (V_{1}, Ψ_{1}, Ω_{1}), G_{2} = (V_{2}, Ψ_{2}, Ω_{2})$ be two FGs. Then the composition of $G_{1}$ and $G_{2}$ is a FG $G_{1} [G_{2}] = (V, Ψ, Ω)$ , where $V = V_{1} \times V_{2}$ , $(u, v), (u_{1}, v_{1}), (u_{2}, v_{2}) \in V, Ψ (u, v) = \land {Ψ_{1} (u), Ψ_{2} (v)}$ and $Ω ((u_{1}, v_{1}), (u_{2}, v_{2})) = {\begin{matrix} \land {Ψ_{1} (u_{1}), Ω_{2} (v_{1}, v_{2})} \\ if u_{1} = u_{2} and v_{1} v_{2} \in E_{2} \\ \land {Ψ_{2} (v_{1}), Ψ_{2} (v_{2}), Ω_{1} (u_{1}, u_{2})} \\ if u_{1} u_{2} \in E_{1} \\ 0 otherwise . \end{matrix}$ Join of two FGs are defined below:

Definition 2.3. [29] Let $G_{1} = (V_{1}, Ψ_{1}, Ω_{1}), G_{2} = (V_{2}, Ψ_{2}, Ω_{2})$ be two FGs. Then the join of $G_{1}$ and $G_{2}$ is a FG $G_{1} + G_{2} = (V, Ψ, Ω)$ , where $V = V_{1} \cup V_{2}$ , $u, v, u_{1}, v_{1} \in V$ , $Ψ (u) = {\begin{matrix} Ψ_{1} (u), if u \in V_{1} \\ Ψ_{2} (u), if u \in V_{2} \end{matrix}$ and $Ω (u, v) = {\begin{matrix} \land {Ψ_{1} (u), Ψ_{2} (v)} if u \in V_{1} and v \in V_{2} \\ Ω_{1} (u, v) if uv \in E_{1} \\ Ω_{2} (u, v) if uv \in E_{2} \\ 0 otherwise . \end{matrix}$ Union of two FGs are defined below:

Definition 2.4. [29] Let $G_{1} = (V_{1}, Ψ_{1}, Ω_{1}), G_{2} = (V_{2}, Ψ_{2}, Ω_{2})$ be two FGs. Then the union of $G_{1}$ and $G_{2}$ is a FG $G_{1} \cup G_{2} = (V, Ψ, Ω)$ , where $V = V_{1} \cup V_{2}$ , $u, v \in V$ , $Ψ (u) = {\begin{matrix} Ψ_{1} (u), if u \in V_{1} \ V_{2} \\ Ψ_{2} (u), if u \in V_{2} \ V_{1} \\ max {Ψ_{1} (u), Ψ_{2} (u)}, if u \in V_{1} \cap V_{2} \end{matrix}$ and $Ω (u, v) = {\begin{matrix} Ω_{1} (u, v) if uv \in E_{1} \ E_{2} \\ Ω_{2} (u, v) if uv \in E_{2} \ E_{1} \\ \lor {Ψ_{1} (u, v), Ψ_{2} (u, v)} if (u, v) \in E_{1} \cap E_{2} \\ 0 otherwise . \end{matrix}$

3 First Zagreb index of fuzzy graphs

Different types of Zagreb index for crisp graphs are shown Table 1. In 2019, Kalathian et al. [22] defined first ZI for a FG as follows:

Table 1
Types of Zagreb index for crisp graph

First Zagreb index $\sum_{v \in V} d^{2} (v)$

Second Zagreb index $\sum_{uv \in E} d (u) d (v)$

Multiplicative first Zagreb index $Π_{uv \in E}$

Multiplicative second Zagreb index $Π_{uv \in E} d (u) d (v)$

Definition 3.1. [22] Let $G = (V, Ψ, Ω)$ be a FG. Then first ZI of the FG $G$ is denoted by $M (G)$ and is defined by: $M (G) = \sum_{v \in V} Ψ (v) d^{2} (v) .$ To maintain the similarity of the first ZI for crisp graphs, in this section, we define the modified ZI of a FG and provide some results on it. Also it is shown that this index gives the better result among other indices for MCDM problems. The definition is given below:

Definition 3.2. Let $G = (V, Ψ, Ω)$ be a FG. Then modified first ZI of the FG $G$ is denoted by ${ZF}_{1} (G)$ and is defined by: ${ZF}_{1} (G) = \sum_{v \in V} [Ψ (v) d (v)]^{2} .$ We call this index the first Zagreb index for fuzzy graphs. First ZI is studied for several fuzzy graphs like path, cycle, star, fuzzy subgraph, etc. and presented many results on it. Also, we have established that the complete fuzzy graph has maximal first Zagreb index among n-vertex fuzzy graphs. Some bounds of first ZI are discussed for Cartesian product, composition, union and join of two fuzzy graphs. An algorithm has been also designed to calculate the first ZI of a fuzzy graph. At the end of the article, a MCDM method is provided using the first Zagreb index of a fuzzy graph to find the best employee in a company. Also a comparison is provided among related indices on the result of application and shown that our method gives better results. An upper bound of first ZI for any FG is given for any FG is given below.

Theorem 3.1. Let $G$ be n-vertex FG with m edges. Then (i) ${ZF}_{1} (G) \leq n^{2} T^{2}$ , (ii) ${ZF}_{1} (G) \leq 4 n^{2} m^{2}$ .

Proof. (i) Using the fact Ψ (v) ≤1, the following inequality holds: $\begin{matrix} {ZF}_{1} (G) & = \sum_{v \in V} [Ψ (v) d (v)]^{2} \leq [\sum_{v \in V} Ψ (v)]^{2} [\sum_{v \in V} d (v)]^{2} \\ \leq n^{2} T^{2} . \end{matrix}$ (ii) Using (i) and the fact $T (G) = 2 \sum_{i = 1}^{m} Ω_{i} \leq 2 m$ , the required inequality follows.

Now, the first ZI at a vertex of a FG is defined below.

Definition 3.3. Let $G$ be a FG. Then first ZI at a vertex v of the FG $G$ is denoted by ${ZF}_{1} (v; G)$ or simply ZF₁ (v) and is defined by ${ZF}_{1} (v) = {ZF}_{1} (G) - {ZF}_{1} (G_{v}) .$

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

Fig. 1

A FG with ${ZF}_{1} (G) = 2.0414$ .

Example 3.1. Let $G$ be a FG shown in Fig. 1 with vertex set $V = {v_{1}, v_{2}, \dots, v_{5}}$ such that, Ψ (v₁) =0.9, Ψ (v₂) =0.5, Ψ (v₃) =0.4, Ψ (v₄) = 0.6, Ψ (v₅) =0.7, Ω (v₁, v₂) =0.3, Ω (v₁, v₄) =0.5, Ω (v₂, v₃) =0.4, Ω (v₃, v₄) =0.4, Ω (v₃, v₅) =0.3, Ω (v₄, v₅) =0.6. Then d (v₁) =0.8, d (v₂) =0.7, d (v₃) =1.1, d (v₄) =1.5, d (v₅) =0.9. Therefore, ${ZF}_{1} (G) = \sum_{v \in V} [Ψ (v) d (v)]^{2} = [0.9 \times 0.8]^{2} + [0.5 \times 0.7]^{2} + [0.4 \times 1.1]^{2} + [0.6 \times 1.5]^{2} + [0.7 \times 0.9]^{2} = 2.0414 .$

The next example shows that the first ZI of a FSG is less than the first ZI of the original FG.

Example 3.2. Let $H$ be a FSG of the FG $G$ shown in Fig. 2 obtained by deletion of the edge v₃v₄. Then d (v₁) =0.8, d (v₂) =0.7, d (v₃) =0.7, d (v₄) =1.1, d (v₅) =0.9. Then, ${ZF}_{1} (H) = 1.5518 \leq {ZF}_{1} (G) .$

Fig. 2

A FSG $H$ of the FG $G$ in the Fig. 1 with ${ZF}_{1} (H) \leq {ZF}_{1} (G)$ .

Proposition 3.1. Let $H = (V^{'}, Ψ^{'}, Ω^{'})$ be a PFSG of a FG $G = (V, Ψ, Ω)$ . Then ${ZF}_{1} (H) \leq {ZF}_{1} (G)$ .

Proof. As $H$ is PFSG of $G$ then for any $u, v \in V^{'}, Ψ^{'} (u) \leq Ψ (u)$ and Ω′ (uv) ≤ Ω (uv). so, $d_{H} (u) = \sum_{v \in V^{'}} Ω^{'} (uv) \leq \sum_{v \in V} Ω (uv) = d_{G} (u) .$ Therefore, ${ZF}_{1} (H) = \sum_{u \in V^{'}} [Ψ^{'} (u) d_{H} (u)]^{2} \leq \sum_{u \in V} [Ψ (u) d_{G} (u)]^{2} = {ZF}_{1} (G) .$ Hence, ${ZF}_{1} (H) \leq {ZF}_{1} (G)$ .

Corollary 3.1. Let $H = (V^{'}, Ψ^{'}, Ω^{'})$ be a FSG of a FG $G = (V, Ψ, Ω)$ . Then ${ZF}_{1} (H) \leq {ZF}_{1} (G)$ . Let 0 ≤ α ≤ 1, the FG $G^{α} = (V^{'}, Ψ^{'}, Ω^{'})$ is a FSG of the FG $G = (V, Ψ, Ω)$ and is defined as $V^{'} = {v \in V : Ψ (v) \leq α}$ and Ψ′ (v) = Ψ (v) , Ω′ (uv) = Ω (uv) for $u, v \in V^{'}$ .

Theorem 3.2. Let $G$ be a FG and let 0 ≤ p₁ ≤ p₂ ≤ 1. Then ${ZF}_{1} (G^{p_{2}}) \leq {ZF}_{1} (G^{p_{1}})$ .

Proof. $G^{p_{2}}$ is PFSG of $G^{p_{1}}$ . Then by Proposition 3, the result follows.

Corollary 3.2. Let $G$ be a FG and let 0 ≤ p₁ ≤ p₂ ≤ ⋯ ≤ p_n ≤ 1. Then ${ZF}_{1} (G^{p_{n}}) \leq {ZF}_{1} (G^{p_{n - 1}}) \leq \dots \leq {ZF}_{1} (G^{p_{2}}) \leq {ZF}_{1} (G^{p_{1}}) .$ In the next theorem, the first ZI of a path is discussed.

Theorem 3.3. Let P (v₀, v₁, ⋯ , v_n) be a path. Then (i) ${ZF}_{1} (P) = Ψ_{0}^{2} Ω_{1}^{2} + Ψ_{n}^{2} Ω_{n}^{2} + \sum_{i = 1}^{n - 1} Ψ_{i}^{2} (Ω_{i} + Ω_{i + 1})^{2},$ (ii) ZF₁ (P) ≤2 (2n - 1) , where Ψ (v_i) = Ψ_i and Ω (v_i-1, v_i) = Ω_i for i = 1, 2, ⋯ , n.

Proof. (i) As P (v₀, v₁, ⋯ , v_n) be a path, then d (v₀) = Ω₁, d (v_n) = Ω_n and d (v_i) = Ω_i + Ω_i+1 for i = 1, 2, ⋯ , n - 1. Therefore, ${ZF}_{1} (P) = Ψ_{0}^{2} Ω_{1}^{2} + \sum_{i = 1}^{n - 1} Ψ_{i}^{2} (Ω_{i} + Ω_{i + 1})^{2} + Ψ_{n}^{2} Ω_{n}^{2} .$ (ii) As Ψ_i ≤ 1, Ω_i ≤ 1, then by using (i) followings are hold: ${ZF}_{1} (P) \leq 1 + \sum_{i = 1}^{n - 1} 2^{2} + 1 = 2 (2 n - 1) .$ First ZI of a cycle is studied in the next theorem.

Theorem 3.4. Let C (v₀, v₁, ⋯ , v_n) be a cycle. Then (i) ${ZF}_{1} (C) = Ψ_{0}^{2} (Ω_{0} + Ω_{n})^{2} + \sum_{i = 1}^{n} Ψ_{i}^{2} (Ω_{i - 1} + Ω_{i})^{2},$ (ii)ZF₁ (C) ≤4 (n + 1) , where Ψ (v_i) = Ψ_i, Ω (v₀, v_n) = Ω_n and Ω (v_i, v_i+1) = Ω_i for i = 0, 1, ⋯ , n - 1. The proof is similar to the proof of Theorem 3.3.

In the next theorem, the first ZI of a star is studied.

Theorem 3.5. Let S (v₀, v₁, ⋯ , v_n) be a fuzzy star. Then (i) ${ZF}_{1} (S) = Ψ_{0}^{2} (\sum_{i = 1}^{n} Ω_{i})^{2} + \sum_{i = 1}^{n} Ψ_{i}^{2} Ω_{i}^{2},$ (ii) ZF₁ (S) ≤ n (n + 1) , where Ψ (v_i) = Ψ_i and Ω (v₀, v_i) = Ω_i for i = 1, 2, ⋯ , n - 1. The proof is similar to the proof of Theorem 3.3.

In the next example, the first ZI of the CFG shown in Fig. 3 is calculated below:

Fig. 3

A CFG with ${ZF}_{1} (G) = 1.0698$ .

Example 3.3. Suppose $G$ be a CFG depicted in Fig. 3 where {v₁, v₂, v₃, v₄} is vertex set and Ψ (v₁) =0.2, Ψ (v₂) =0.4, Ψ (v₃) =0.5, Ψ (v₄) =0.7. Then d (v₁) =0.6, d (v₂) =1, d (v₃) =1.1, d (v₄) =1.1. Therefore, ${ZF}_{1} (G) = \sum_{v \in V} [Ψ (v) d (v)]^{2} = [0.2 \times 0.6]^{2} + [0.4 \times 1]^{2} + [0.5 \times 1.1]^{2} + [0.7 \times 1.1]^{2} = 1.0698 .$ First ZI of a CFG is studied in the next theorem.

Theorem 3.6. Let $G = (V, Ψ, Ω)$ be a CFG with vertex set $V = {v_{1}, v_{2}, \dots, v_{n}}$ . Then (i) ${ZF}_{1} (G) = \sum_{i = 1}^{n} Ψ_{i}^{2} [\sum_{j = 1}^{i - 1} Ψ_{j} + (n - i) Ψ_{i}]^{2}$ , (ii) $n (n - 1)^{2} Ψ_{0}^{4} \leq {ZF}_{1} (G) \leq n (n - 1)^{2} Ψ_{n}^{4} \leq n (n - 1)^{2}$ , where Ψ (v_i) = Ψ_i and Ψ₁ ≤ Ψ₂ ≤ ⋯ ≤ Ψ_n for i = 1, 2, ⋯ , n.

Proof. (i) As $G$ be a CFG, then $d (v_{i}) = (n - i) Ψ_{i} + \sum_{j = 1}^{i - 1} Ψ_{j}$ . Therefore, ${ZF}_{1} (G) = \sum_{i = 1}^{n} Ψ_{i}^{2} [(n - i) Ψ_{i} + \sum_{j = 1}^{i - 1} Ψ_{j}]^{2} .$ (ii) From (i) and Ψ₀ ≤ Ψ_i ≤ Ψ_n, the following inequalities holds: ${ZF}_{1} (G) \leq \sum_{i = 1}^{n} Ψ_{n}^{2} [(n - i) Ψ_{n} + \sum_{j = 1}^{i - 1} Ψ_{n}]^{2} = Ψ_{n}^{4} \sum_{i = 1}^{n} [n - i + i - 1]^{2} = n (n - 1)^{2} Ψ_{n}^{4} .$ Other inequalities follow similarly. Let $G = (V, Ψ, Ω)$ be a FG. The FG $C [G] = (V, Ψ, Ω^{c})$ is constructed as Ω^c (uv) = ∧ {Ψ (u) , Ψ (v)} and we called the FG is completion fuzzy graph of the FG $G$ .

Theorem 3.7. Let $G$ be a FG. Then ${ZF}_{1} (G) \leq {ZF}_{1} (C [G])$ .

Proof. As $C [G]$ is completion fuzzy graph of the FG $G$ , then for any $(uv) \in E, Ω (uv) \leq Ω^{c} (uv)$ . So, $G$ is PFSG of $C [G]$ . Then by Proposition 3, the result follows.

Corollary 3.3. For any n-vertex FG $G$ , ${ZF}_{1} (G) \leq n (n - 1)^{2}$ . In the next theorem, first ZI is discussed for isomorphic FGs.

Theorem 3.8. Let $G_{1}$ and $G_{2}$ be isomorphic. Then ${ZF}_{1} (G_{1}) = {ZF}_{1} (G_{2})$ .

Proof. As $G_{1}$ and $G_{2}$ are isomorphic FGs, there exist a mapping $Φ : V_{1} \to V_{2}$ which is a bijection and for all $u, v \in V_{1}, Ψ_{1} (v) = Ψ_{2} (Φ (v))$ and Ω₁ (uv) = Ω₂ (Φ (u) , Φ (v)). Then, $d_{G_{1}} (v) = \sum_{u \in V_{1}} Ω_{2} (Φ (u), Φ (v))$ $= \sum_{Φ (u) \in V_{2}} Ω_{2} (Φ (u), Φ (v)) = d_{G_{2}} (Φ (v)) .$ Therefore, ${ZF}_{1} (G_{1}) = \sum_{v \in V_{1}} [Ψ_{2} (Φ (v)) d_{G_{2}} (Φ (v))]^{2}$ $= \sum_{Φ (v) \in V_{2}} [Ψ_{2} (φ (v)) d_{G_{2}} (Φ (v))]^{2} = {ZF}_{1} (G_{2}) .$ This completes the proof.

The bounds for the first ZI of any FG are calculated below.

Theorem 3.9. Let $G$ be a n-vertex FG with size m. Then $\frac{n δ^{4}}{m^{2}} \leq {ZF}_{1} (G) \leq n Δ^{2}$ .

Proof. Now for $v \in V, δ \leq d (v) \leq m Ψ (v)$ . Therefore, $Ψ (v) \geq \frac{δ}{m}$ . Then, ${ZF}_{1} (G) = \sum_{v \in V} [Ψ (v) d (v)]^{2}$ $\leq Δ^{2} \sum_{v \in V} [Ψ (v)]^{2} \leq n Δ^{2} .$ Again, ${ZF}_{1} (G) = \sum_{v \in V} [Ψ (v) d (v)]^{2}$ $\geq δ^{2} \sum_{v \in V} [Ψ (v)]^{2} \geq δ^{2} \sum_{1}^{n} (\frac{δ}{m})^{2} = \frac{n δ^{4}}{m^{2}} .$ This completes the proof.

4 Bounds of first Zagreb index of fuzzy graphs during operations

In this section, some bounds of first ZI are discussed during FG operations.

Theorem 4.1. ${ZF}_{1} (G_{1} \times G_{2}) \leq n_{2} {ZF}_{1} (G_{1}) + n_{1} {ZF}_{1} (G_{2}) + 2 n_{1} n_{2} Δ_{1} Δ_{2}$ .

Proof. As $G_{1} \times G_{2}$ is CPG of $G_{1}$ and $G_{2}$ , then for $(u, v), (u_{1}, v_{1}) \in V, Ψ (u, v) = \land {Ψ_{1} (u), Ψ_{2} (v)}$ and $Ω ((u, v) (u_{1}, v_{1})) = {\begin{matrix} \land {Ψ_{1} (u), Ω_{2} ({vv}_{1})} if u = u_{1}, \\ v v_{1} \in E_{2} \\ \land {Ψ_{2} (v), Ω_{1} ({uu}_{1})} if u u_{1} \in E_{1}, \\ v = v_{1} \\ 0 otherwise . \end{matrix}$ $Then d_{G_{1} \times G_{2}} (u, v) = \sum_{v_{i} v \in E_{2}} Ω {(u, v) (u, v_{i})} + \sum_{u_{i} u \in E_{1}} Ω {(u, v) (u_{i}, v)} = \sum_{v_{i} v \in E_{2}} \land {Ψ_{1} (u), Ω_{2} ({vv}_{i})}$ $+ \sum_{u_{i} u \in E_{1}} \land {Ψ_{2} (v), Ω_{1} ({uu}_{i})} \leq \sum_{v_{i} v \in E_{2}} Ω_{2} ({vv}_{i}) + \sum_{u_{i} u \in E_{1}} Ω_{1} ({uu}_{i}) = d_{G_{1}} (u) + d_{G_{2}} (v) .$ Now first ZI of $G_{1} \times G_{2}$ is: $\begin{matrix} {ZF}_{1} (G_{1} \times G_{2}) = \sum_{(u, v) \in V} [Ψ (u, v) d_{G_{1} \times G_{2}} (u, v)]^{2} \\ \leq \sum_{(u, v) \in V} [\land {Ψ_{1} (u), Ψ_{2} (v)} {d_{G_{1}} (u) + d_{G_{2}} (v)}]^{2} \\ \leq \sum_{(u, v) \in V} [Ψ_{1} (u) d_{G_{1}} (u) + Ψ_{2} (v) d_{G_{2}} (v)]^{2} \\ = \sum_{(u, v) \in V} [Ψ_{1} (u) d_{G_{1}} (u)]^{2} + \sum_{(u, v) \in V} [Ψ_{2} (v) d_{G_{2}} (v)]^{2} \\ + 2 \sum_{(u, v) \in V} Ψ_{1} (u) d_{G_{1}} (u) Ψ_{2} (v) d_{G_{2}} (v) \\ \leq \sum_{v \in V_{2}} {ZF}_{1} (G_{1}) + \sum_{u \in V_{1}} {ZF}_{1} (G_{2}) + 2 \sum_{(u, v) \in V} Δ_{1} Δ_{2} \\ = n_{2} {ZF}_{1} (G_{1}) + n_{1} {ZF}_{1} (G_{2}) + 2 n_{1} n_{2} Δ_{1} Δ_{2} . \end{matrix}$

Corollary 4.1. (i) ${ZF}_{1} (G_{1} \times G_{2}) \leq n_{1} n_{2} (Δ_{1} + Δ_{2})^{2},$ (ii) ${ZF}_{1} (G_{1} \times G_{2}) \leq n_{1} n_{2} (n_{1} + n_{2} - 2)^{2},$ (iii) ${ZF}_{1} (G_{1} \times G_{2}) \leq 4 m_{1}^{2} n_{1}^{2} n_{2} + 4 m_{2}^{2} n_{2}^{2} n_{1} + 2 n_{1} n_{2} (n_{1} - 1) (n_{2} - 1)$ .

Proof. (i) From Theorem 4.1 and 3.9, the following inequalities are holds: ${ZF}_{1} (G_{1} \times G_{2}) \leq n_{1} n_{2} Δ_{1}^{2} + n_{1} n_{2} Δ_{2}^{2} + 2 n_{1} n_{2} Δ_{1} Δ_{2} = n_{1} n_{2} (Δ_{1} + Δ_{2})^{2} .$ (ii) Using (i) and the facts Δ₁ ≤ n₁ - 1 and Δ₂ ≤ n₂ - 1, the result follows. (iii) Using the inequalities in Theorem 4.1, 3.1 and the facts Δ₁ ≤ n₁ - 1 and Δ₂ ≤ n₂ - 1, the inequality hold.

Theorem 4.2. ${ZF}_{1} (G_{1} [G_{2}]) \leq n_{2}^{3} {ZF}_{1} (G_{1}) + n_{1} {ZF}_{1} (G_{2}) + 2 n_{1} n_{2}^{2} Δ_{1} Δ_{2}$ .

Proof. As $G_{1} [G_{2}]$ is composition graph of $G_{1}$ and $G_{2}$ , then for $(u, v), (u_{1}, v_{1}) \in V, Ψ (u, v) = \land {Ψ_{1} (u), Ψ_{2} (v)}$ and $Ω ((u, v) (u_{1}, v_{1})) = {\begin{matrix} \land {Ψ_{1} (u), Ω_{2} ({vv}_{1})} \\ if u = u_{1}, {vv}_{1} \in E_{2} \\ \land {Ψ_{2} (v), Ψ_{2} (v_{1}), Ω_{1} ({uu}_{1})} \\ if {uu}_{1} \in E_{1} \\ 0 otherwise . \end{matrix}$ Then $d_{G_{1} [G_{2}]} (u, v) = \sum_{v_{i} v \in E_{2}} Ω {(u, v) (u, v_{i})} + \sum_{u_{i} u \in E_{1}, v_{j} \in V_{2}} Ω {(u, v) (u_{i}, v_{j})}$ $= \sum_{v_{i} v \in E_{2}} \land {Ψ_{1} (u), Ω_{2} ({vv}_{i})}$ $+ \sum_{u_{i} u \in E_{1}, v_{j} \in V_{2}} \land {Ψ_{2} (v), Ψ_{2} (v_{j}) Ω_{1} ({uu}_{i})} \leq$ $\sum_{v_{i} v \in E_{2}} Ω_{2} ({vv}_{i}) + \sum_{u_{i} u \in E_{1}, v_{j} \in V_{2}} Ω_{1} ({uu}_{i}) =$ $n_{2} d_{G_{1}} (u) + d_{G_{2}} (v) .$ Then first ZI of $G_{1} [G_{2}]$ is: $\begin{matrix} {ZF}_{1} (G_{1} [G_{2}]) = \sum_{(u, v) \in V} [Ψ (u, v) d_{G_{1} [G_{2}]} (u, v)]^{2} \\ \leq \sum_{(u, v) \in V} [\land {Ψ_{1} (u), Ψ_{2} (v)} {n_{2} d_{G_{1}} (u) + d_{G_{2}} (v)}]^{2} \\ \leq \sum_{(u, v) \in V} [n_{2} Ψ_{1} (u) d_{G_{1}} (u) + Ψ_{2} (v) d_{G_{2}} (v)]^{2} \\ = n_{2}^{2} \sum_{(u, v) \in V} [Ψ_{1} (u) d_{G_{1}} (u)]^{2} + \sum_{(u, v) \in V} [Ψ_{2} (v) d_{G_{2}} (v)]^{2} \\ + 2 n_{2} \sum_{(u, v) \in V} Ψ_{1} (u) d_{G_{1}} (u) Ψ_{2} (v) d_{G_{2}} (v) \\ \leq n_{2}^{2} \sum_{v \in V_{2}} {ZF}_{1} (G_{1}) + \sum_{u \in V_{1}} {ZF}_{1} (G_{2}) + 2 n_{2} \sum_{(u, v) \in V} Δ_{1} Δ_{2} \\ = n_{2}^{3} {ZF}_{1} (G_{1}) + n_{1} {ZF}_{1} (G_{2}) + 2 n_{1} n_{2}^{2} Δ_{1} Δ_{2} . \end{matrix}$

Corollary 4.2. (i) ${ZF}_{1} (G_{1} [G_{2}]) \leq n_{2}^{3} (n_{1}^{3} + 2 n_{1}^{2} + 1)$ and (ii) ${ZF}_{1} (G_{1} [G_{2}]) \leq 4 m_{1}^{2} n_{1}^{2} n_{2}^{3} + 4 m_{2}^{2} n_{2}^{2} + 2 n_{1} n_{2}^{2} (n_{1} - 1) (n_{2} - 1)$ .

Theorem 4.3. ${ZF}_{1} (G_{1}) + {ZF}_{1} (G_{2}) \leq {ZF}_{1} (G_{1} + G_{2}) \leq {ZF}_{1} (G_{1}) + {ZF}_{1} (G_{2}) + n_{1} n_{2} (n_{1} + n_{2}) + 2 n_{1} n_{2} (Δ_{1} + Δ_{2})$ .

Proof. As $G_{1} + G_{2}$ is join of $G_{1}$ and $G_{2}$ , then for $u, v \in V$ , $Ψ (u) = {\begin{matrix} Ψ_{1} (u), if u \in V_{1} \\ Ψ_{2} (u), otherwise \end{matrix}$ $Ω (uv) = {\begin{matrix} \land {Ψ_{1} (u), Ψ_{2} (v)} if u \in V_{1}, v \in V_{2} \\ Ω_{1} (uv) if uv \in E_{1} \\ Ω_{2} (uv) if uv \in E_{2} \\ 0 otherwise . \end{matrix}$ Then the degree of the vertex u is given below: $d_{G_{1} + G_{2}} (u) = {\begin{matrix} d_{G_{1}} (u) + \sum_{v \in V_{2}} \land {Ψ_{1} (u), Ψ_{2} (v)}, \\ u \in V_{1} \\ d_{G_{2}} (u) + \sum_{v \in V_{1}} \land {Ψ_{2} (u), Ψ_{1} (v)}, \\ u \in V_{2} \end{matrix} \geq {\begin{matrix} d_{G_{1}} (u), u \in V_{1} \\ d_{G_{2}} (u), u \in V_{2} . \end{matrix}$ Again, from equation (1), the following hold: $d_{G_{1} + G_{2}} (u) \leq {\begin{matrix} d_{G_{1}} (u) + \sum_{v \in V_{2}} Ψ_{1} (u), u \in V_{1} \\ d_{G_{2}} (u) + \sum_{v \in V_{1}} Ψ_{2} (u), u \in V_{2} \end{matrix} \leq {\begin{matrix} d_{G_{1}} (u) + n_{2} Ψ_{1} (u), u \in V_{1} \\ d_{G_{2}} (u) + n_{1} Ψ_{2} (u), u \in V_{2} \end{matrix}$ Then first ZI of $G_{1} + G_{2}$ is: $\begin{matrix} {ZF}_{1} (G_{1} + G_{2}) = \sum_{u \in V} [Ψ (u) d_{G_{1} + G_{2}} (u)]^{2} \\ = \sum_{u \in V_{1}} [Ψ_{1} (u) d_{G_{1} + G_{2}} (u)]^{2} + \sum_{u \in V_{2}} [Ψ_{2} (u) d_{G_{1} + G_{2}} (u)]^{2} \\ \geq \sum_{u \in V_{1}} [Ψ_{1} (u) d_{G_{1}} (u)]^{2} + \sum_{u \in V_{2}} [Ψ_{2} (u) d_{G_{2}} (u)]^{2} \\ = {ZF}_{1} (G_{1}) + {ZF}_{2} (G_{2}) . \end{matrix}$ Now, $\begin{matrix} {ZF}_{1} (G_{1} + G_{2}) = \sum_{u \in V} [Ψ (u) d_{G_{1} + G_{2}} (u)]^{2} \\ = \sum_{u \in V_{1}} [Ψ_{1} (u) d_{G_{1} + G_{2}} (u)]^{2} + \sum_{u \in V_{2}} [Ψ_{2} (u) d_{G_{1} + G_{2}} (u)]^{2} \\ \leq \sum_{u \in V_{1}} [Ψ_{1} (u) {d_{G_{1}} (u) + n_{2} Ψ_{1} (u)}]^{2} \\ + \sum_{u \in V_{2}} [Ψ_{2} (u) {d_{G_{2}} (u) + n_{2} Ψ_{2} (u)}]^{2} \\ = K_{1} + K_{2}, \end{matrix}$ Where $K_{1} = \sum_{u \in V_{1}} [Ψ_{1} (u) {d_{G_{1}} (u) + n_{2} Ψ_{1} (u)}]^{2}$ and $K_{2} = \sum_{u \in V_{2}} [Ψ_{2} (u) {d_{G_{2}} (u) + n_{2} Ψ_{2} (u)}]^{2}$ Now $K_{1} = \sum_{u \in V_{1}} [Ψ_{1} (u) {d_{G_{1}} (u) + n_{2} Ψ_{1} (u)}]^{2} = \sum_{u \in V_{1}} [Ψ_{1} (u) d_{G_{1}} (u)]^{2} + n_{2}^{2} \sum_{u \in V_{1}} [Ψ_{1} (u)]^{4} + 2 n_{2} \sum_{u \in V_{1}} Ψ_{1}^{2} (u) d_{G_{1}} (u) \leq {ZF}_{1} (G_{1}) + n_{1} n_{2}^{2} + 2 n_{1} n_{2} Δ_{1} .$ Similarly, $K_{2} \leq {ZF}_{1} (G_{2}) + n_{2} n_{1}^{2} + 2 n_{1} n_{2} Δ_{2}$ .

Hence the result follows.

Corollary 4.3. (i) ${ZF}_{1} (G_{1} + G_{2}) \leq (n_{1} + n_{2}) (n_{1} + n_{2} - 1)^{2}$ and (ii) ${ZF}_{1} (G_{1} + G_{2}) \leq 4 m_{1}^{2} n_{1}^{2} + 4 m_{2}^{2} n_{2}^{2} + n_{1} n_{2} (3 n_{1} + 3 n_{2} - 4)$ .

Theorem 4.4. ${ZF}_{1} (G_{1} \cup G_{2}) \geq {ZF}_{1} (G_{1}) + {ZF}_{1} (G_{2}) - k Δ^{2}$ , where k = |V₁ ∩ V₂|, Δ = max {Δ₁, Δ₂}.

Proof. As $G_{1} \cup G_{2}$ is union of $G_{1}$ and $G_{2}$ , then for $u, v \in V$ $Ψ (u) = {\begin{matrix} Ψ_{1} (u), if u \in V_{1} \ V_{2} \\ Ψ_{2} (u), if u \in V_{2} \ V_{1} \\ \lor {Ψ_{1} (u), Ψ_{2} (u)}, if u \in V_{1} \cap V_{2} \end{matrix}$ and $Ω (u, v) = {\begin{matrix} Ω_{1} (u, v) if uv \in E_{1} \ E_{2} \\ Ω_{2} (u, v) if uv \in E_{2} \ E_{1} \\ \lor {Ψ_{1} (u, v), Ψ_{2} (u, v)} \\ if (u, v) \in E_{1} \cap E_{2} \\ 0 otherwise . \end{matrix}$

Then the degree of a vertex u is given below: $d_{G_{1} \cup G_{2}} (u) \geq {\begin{matrix} d_{1} (u) if u \in V_{1} \ V_{2} \\ d_{2} (u) if u \in V_{2} \ V_{1} \\ d_{1} (u) or d_{2} (u) if u \in V_{1} \cap V_{2} \end{matrix}$ $\begin{matrix} Now, & {ZF}_{1} (G_{1} \cup G_{2}) = \sum_{u \in V_{1}} [Ψ_{1} (u) d_{1} (u)]^{2} \\ + \sum_{u \in V_{2}} [Ψ_{2} (u) d_{2} (u)]^{2} - \sum_{u \in V_{1} \cap V_{2}} [Ψ (u) d (u)]^{2} \\ \geq {ZF}_{1} (G_{1}) + {ZF}_{1} (G_{2}) - k Δ^{2} . \end{matrix}$

5 Algorithm to compute first Zagreb index of a fuzzy graph

In this section, we present an algorithm to calculate the first ZI of a FG. The algorithm is given below: Input: A FG $G = (V, Ψ, Ω)$ . Output: First ZI of the FG $G$ and of any vertex of the FG. Step 1: Evaluate the degree of every vertex in $G$ as $d (u) = \sum_{v \in V} Ω (u, v)$ . Step 2: Construct the FSG $G_{v}$ for each $v \in V$ . Step 3: Calculate the first ZI of the FG $G$ as ${ZF}_{1} (G) = \sum_{v \in V} [Ψ (v) d (v)]^{2} .$ Step 4: Calculate the first ZI of a vertex v of the FG as ${ZF}_{1} (v) = {ZF}_{1} (G) - {ZF}_{1} (G_{v}) .$ Step 5: End.

6 Model for MCDM problem using first Zagreb index of a fuzzy graph

There are many real life problems which can be described by FGs. In this section, a MCDM problem is presented and solved by first ZI. For various MCDM problems, one can see [8 , 16–20]. In the study of human group behavior in a network, it is noticed that individual ideas have considerable influence on others. First ZI of a FG can be used to model such behavior. Let A = {A₁, A₂, ⋯ , A_n} be the set of alternatives and B = {B₁, B₂, ⋯ , B_m} be the set of attributes, W = {w₁, w₂, ⋯ , w_m} be the weight vector of the attributes, where $w_{j} \in (0, 1), \sum_{j = 1}^{m} w_{j} = 1$ . Now our aim is to find the best alternative. To find the best alternative, first we construct a FGs for each attribute whose vertex set is the set of alternatives A and edges are the influence relation between the alternatives. Then the following algorithm is performed to select the most appropriate alternatives. The algorithm is given below:

Input: The FG G_{B
_i} corresponding to the attribute B_i with vertex set is the set of alternatives A and edge represents the influence relation between the vertices, for each i = 1, 2, ⋯ , m.

Output: Best alternatives.

Step 1: Compute the first ZI for each vertex of every FGs by using the algorithm presented in the section 5.

Step 2: Calculate the score of each vertex by using the formula $S (A_{j}) = \sum_{i = 1}^{m} w_{i} ZF (A_{j}; G_{e_{i}})$

Step 3: Calculate the normalized score of each vertex by using the formula $NS (A_{j}) = \frac{S (A_{j})}{\sum_{i = 1}^{m} w_{i} ZF (G)}$

Step 4: Compare A_i, A_j by NS (A_i) , NS (A_j). That is if NS (A_i) > NS (A_j) then A_i ≻ A_j and we call it A_i is better than A_j.

Step 5: Find the best alternative A_k for some k = 1, 2, ⋯ , n where A_k ≻ A_i for all i = 1, 2, ⋯ , k - 1, k + 1, k + 2, ⋯ , n.

6.1 A case study

In this part, we shall present a MCDM problem to show the application of the proposed method. Suppose a company wants to know who is the most valuable employee among all workers in the company. Let, the employees (alternatives) A, B, C, D and the parameters (attributes) which used to evaluate their quality-value be P₁, P₂, P₃, P₄, P₅, P₆. The parameters are defined as P₁ = hard working, P₂ = ambitious, P₃ = creative, P₄ = team spirit, P₅ = cooperative, P₆ = committed. These parameters need not be the same for all employees. But, all the parameters have their importance to evaluate their performance. Since these parameters are linguesting terms, there are no fixed values for them. These terms can be mathematically represented by assigning membership values corresponding to each parameter. The employees and parameters can be represented by a FG. For this, we need to develop a FGs corresponding to each parameter whose vertex set is the set of employees and the edge represents the influence relation between the vertices. Suppose the company provided the score value of each employee and score value of the influence relation between the employees for each parameter in the Table 2 and 3 respectively. Note that all score values are calculated out of 1 by the company. From the Table 2 and 3 one can construct several graphs G_{P
_i} shown in Fig. 4. The FG, G_{P
_i} is constructed whose vertex set is the set of employees A, B, C, D. The MV of a vertex is the score value of corresponding employee (taken from Table 2) and there is an edge between two (employees) vertices if the score value of the influence relation between the employees is positive (shown in the Table 3) and this score value is the MV of that edge. The weights of the parameters P₁, P₂, P₃, P₄, P₅ and P₆ are taken as 0.3, 0.2, 0.2, 0.1, 0.1, 0.1 respectively.

Table 2
Score value of employees (given by the company)

Employee P ₁ P ₂ P ₃ P ₄ P ₅ P ₆

A 0.5 0.6 0.7 0.4 0.9 0.4

B 0.4 0.3 0.6 0.7 0.7 0.5

C 0.6 0.5 0.8 0.5 0.5 0.6

D 0.7 0.7 0.5 0.6 0.6 0.5

Employee	P ₁	P ₂	P ₃	P ₄	P ₅	P ₆
A	0.5	0.6	0.7	0.4	0.9	0.4
B	0.4	0.3	0.6	0.7	0.7	0.5
C	0.6	0.5	0.8	0.5	0.5	0.6
D	0.7	0.7	0.5	0.6	0.6	0.5

Table 3

Score value of the influence relation between the employees (given by the company)

P ₁	A	B	C	D	P ₂	A	B	C	D	P ₃	A	B	C	D
A		0.4	0.0	0.3	A		0.3	0.5	0.5	A		0.6	0.0	0.0
B	0.4		0.3	0.4	B	0.3		0.3	0.0	B	0.6		0.5	0.4
C	0.0	0.3		0.4	C	0.5	0.3		0.4	C	0.0	0.5		0.5
D	0.3	0.4	0.4		D	0.5	0.0	0.4		D	0.0	0.4	0.5
P ₁	A	B	C	D	P ₂	A	B	C	D	P ₃	A	B	C	D
A		0.4	0.4	0.3	A		0.7	0.4	0.6	A		0.6	0.0	0.0
B	0.4		0.0	0.6	B	0.7		0.3	0.5	B	0.6		0.5	0.4
C	0.4	0.0		0.4	C	0.4	0.3		0.0	C	0.0	0.5		0.5
D	0.3	0.6	0.4		D	0.6	0.5	0.0		D	0.0	0.4	0.5

Fig. 4

FG representation of the MCDM problem.

Solve by using first Zagreb index

Step 1: Using the algorithm presented in Section 5, first ZI of each employee are given in the Table 4.

Table 4

First ZI of the employee corresponding to the graph in Fig. 4.

ZF ₁	G _{P ₁}	G _{P ₂}	G _{P ₃}	G _{P ₄}	G _{P ₅}	G _{P ₆}
ZF₁ (A)	0.5170	1.1887	0.6948	0.8756	3.5754	0.3409
ZF₁ (B)	0.7652	0.4383	1.6064	1.0372	3.0219	0.4989
ZF₁ (C)	0.6204	1.0367	1.2525	0.5920	1.4914	0.1749
ZF₁ (D)	0.9346	0.9749	1.0569	1.2312	2.4089	0.3553

Step 2: Then the score of each employee is calculated and shown in the Table 5.

Table 5

Score and normalized score of the employees.

	A	B	C	D
Score	1.01099	1.09430	0.86979	1.08628
Normalized Score	0.599066	0.648432	0.515389	0.64368

Step 3: Then the normalized score of the each employee is calculated and shown in the Table 5.

Step 4: So the order of the alternatives is B ≻ D ≻ A ≻ C.

Step 5: Therefore the B is the most valuable employee.

7 Analysis of first Zagreb index and its comparison with existing indices on result of application

For this comparison, we considered the MCDM problem described in the Subsection 6.1 and compared with first Zagreb index, second Zagreb index, multiplicative first Zagreb index and multiplicative second Zagreb index of a crisp graph. It is noticed that other indices depend only on the number of neighbours of each employee but not depends on the nature of the neighbour. Here the first Zagreb index of fuzzy graphs not only depends on the degree of a vertex but also depends on nature of the vertex and nature of the neighbours. So this index for MCDM problems always shows a realistic result compared to other existing indices. The score and normalized score of each employee for those indices are shown in Table 6. The comparison table of those normalized indices are presented in Table 7. It is observed that each index provides a similar decision. But, from Table 7, It is seen that other indices are giving higher normalized score values compared to our index. It is also observed that the difference between normalized scores of each employee are very small for normalized multiplicative first Zagreb index and normalized multiplicative second Zagreb index of crisp graph but our index provides the higher differences.

Table 6
Score and Normalized Score of Employees

Employee Name First Zagreb index of fuzzy graph Normalized First Zagreb index of fuzzy graph First Zagreb index of crisp graph Normalized First Zagreb index of crisp graph Second Zagreb index of crisp graph Normalized Second Zagreb index of crisp graph Multiplicative First Zagreb index of crisp graph Normalized Multiplicative First Zagreb index of crisp graph Multiplicative Second Zagreb index of crisp graph Normalized Multiplicative Second Zagreb index of crisp graph

A 1.011 0.599066 14.6 0.618644 20.8 0.722222 2708.5 0.986703 8260.4 0.995901

B 1.0943 0.648432 17 0.720339 23.3 0.809028 2721.9 0.991585 8273.3 0.997456

C 0.8698 0.515389 14 0.59322 20 0.694444 2703 0.984699 8254.4 0.995177

D 1.0863 0.64368 15.8 0.669492 22.4 0.777778 2719.5 0.99071 8278.4 0.998071

Employee Name	First Zagreb index of fuzzy graph	Normalized First Zagreb index of fuzzy graph	First Zagreb index of crisp graph	Normalized First Zagreb index of crisp graph	Second Zagreb index of crisp graph	Normalized Second Zagreb index of crisp graph	Multiplicative First Zagreb index of crisp graph	Normalized Multiplicative First Zagreb index of crisp graph	Multiplicative Second Zagreb index of crisp graph	Normalized Multiplicative Second Zagreb index of crisp graph
A	1.011	0.599066	14.6	0.618644	20.8	0.722222	2708.5	0.986703	8260.4	0.995901
B	1.0943	0.648432	17	0.720339	23.3	0.809028	2721.9	0.991585	8273.3	0.997456
C	0.8698	0.515389	14	0.59322	20	0.694444	2703	0.984699	8254.4	0.995177
D	1.0863	0.64368	15.8	0.669492	22.4	0.777778	2719.5	0.99071	8278.4	0.998071

Table 7

Comparison Table

8 Limitations of the paper

The limitations of the paper are: (i) Good upper bound of the first Zagreb index for path, cycle, star, complete fuzzy graph, etc. are presented here but we cannot provide the good lower bound. (ii) In this paper, it is shown that for n-vertex fuzzy graphs, the complete fuzzy graph has a maximum first Zagreb index. But for which n-vertex fuzzy graph has the minimum first Zagreb index? (iii) which n-vertex tree (fuzzy) has the maximum first Zagreb index? (iv) which n-vertex tree (fuzzy) has the minimum first Zagreb index? (v) In [21] Khalifeh et al. has given the exact expressions of the first Zagreb index for crisp graph operations (Cartesian product, composition, join, union). But for fuzzy graph operations, we cannot provide the exact values of the first Zagreb index.

9 Conclusion

Zagreb index has an important role in spectral graph theory, chemical graph theory, biochemistry, etc. In this article, the first Zagreb for fuzzy graphs is discussed and provides some upper bound of such TIs for path, star, complete fuzzy graph etc. In this article, it is shown that a complete fuzzy graph is the maximal fuzzy graph with respect to first ZIs for a given vertex set. Some bounds of first ZIs are established for Cartesian product, composition, union, join of two FGs. An algorithm is also designed to compute the first ZI of a FG. At the end, a MCDM method is provided using the first Zagreb index of a fuzzy graph to find the best employee in a company. Also a comparison is provided among related indices on the result of application and shown that our method gives better results. One can introduce the second Zagreb index and other TIs for fuzzy graphs and establish those types of results also.

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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First Zagreb index	$\sum_{v \in V} d^{2} (v)$
Second Zagreb index	$\sum_{uv \in E} d (u) d (v)$
Multiplicative first Zagreb index	$Π_{uv \in E}$
Multiplicative second Zagreb index	$Π_{uv \in E} d (u) d (v)$

First Zagreb index on a fuzzy graph and its application

Abstract

Keywords

Abbreviations

1 Introduction

1.1 Research background

1.2 Research question

1.3 Objective of the work

1.4 Structure of the study

2 Preliminaries

3 First Zagreb index of fuzzy graphs

Table 1 Types of Zagreb index for crisp graph First Zagreb index ∑ v ∈ V d 2 ( v ) Second Zagreb index ∑ uv ∈ E d ( u ) d ( v ) Multiplicative first Zagreb index Π uv ∈ E Multiplicative second Zagreb index Π uv ∈ E d ( u ) d ( v )

5 Algorithm to compute first Zagreb index of a fuzzy graph

6 Model for MCDM problem using first Zagreb index of a fuzzy graph

6.1 A case study

Table 2 Score value of employees (given by the company) Employee P 1 P 2 P 3 P 4 P 5 P 6 A 0.5 0.6 0.7 0.4 0.9 0.4 B 0.4 0.3 0.6 0.7 0.7 0.5 C 0.6 0.5 0.8 0.5 0.5 0.6 D 0.7 0.7 0.5 0.6 0.6 0.5

9 Conclusion

Footnotes

Acknowledgments

Compliance with ethical standards

References

Table 1
Types of Zagreb index for crisp graph

First Zagreb index $\sum_{v \in V} d^{2} (v)$

Second Zagreb index $\sum_{uv \in E} d (u) d (v)$

Multiplicative first Zagreb index $Π_{uv \in E}$

Multiplicative second Zagreb index $Π_{uv \in E} d (u) d (v)$

Table 2
Score value of employees (given by the company)

Employee P ₁ P ₂ P ₃ P ₄ P ₅ P ₆

A 0.5 0.6 0.7 0.4 0.9 0.4

B 0.4 0.3 0.6 0.7 0.7 0.5

C 0.6 0.5 0.8 0.5 0.5 0.6

D 0.7 0.7 0.5 0.6 0.6 0.5