Introduction

Research gap and motivation

In 1965, Zadeh [39] first introduced the idea of uncertainty in a classical set and the new set is called by “Fuzzy set”. Jafar et al. studied distance and similarity measures of neutrosophic hypersoft environment [15], matrix theory for neutrosophic hypersoft environment [11], interval valued fuzzy soft environment [12], trigonometric similarity measures for neutrosophic hypersoft environment [14], similarity measures of tangent, cotangent and cosines in neutrosophic environment [13]. Kumar et al. studied type-1 to type-4 transportation problem [17, 18], intuitionistic fuzzy 3D assignment problem [21], optimization problem in fuzzy and intuitionistic fuzzy sets [20], intuitionistic fuzzy solid assignment problem [19], fully intuitionistic fuzzy transportation problem [17].

Rosenfeld [35] introduced the fuzzy graph (FG) in 1975. Samanta et al. [36] have discussed the fuzzy colouring of FGs. Mahapatra et al. has introduced the radio FG [23], edge colouring of FGs [24]. Rashmanlou et al. has introduced bipolar FGs [32], categorical properties of bipolar FGs [33], intuitionistic FGs [34]. Samanta and Pal [37] has introduced k-competition and p-competition FGs.

Topological indices are evaluated on a molecular graph for a chemical compound that describes the compound’s topology. An atom is considered a vertex and a bond is taken as an edge for a chemical compound. During the research on boiling point of paraffins, Wiener [38] first introduced a topological index: Wiener index (WI) in 1947. After that, Gutman and Trinajstic [4] has introduced the Zagreb index (ZI) and applied it to determine the energy of \(\pi \)-electron. F-index has introduced by Fortula and Gutman [3] in 2015. Neighbourhood-ZI [26] is introduced by Mondal et al. They also studied Neighbourhood-ZI in QSPR research [25].

Recently, Wiener index (WI) [2], connectivity index (CI) [1] of a FG is introduced by Binu et al. Islam et al. also discussed the WI [10] for a saturated FGs. Islam and Pal have also developed the concept of hyper-WI [8], hyper-CI [6], first ZI [7] and F-index [5, 9] in FGs. The Wiener absolute index [30], certain index [29] and Randic index [31] for bipolar FGs are introduced by Poulik et al. For many other topological indices for FGs one can see [16]. Motivated these article, here we have introduced hyper-ZI for FGs.

Main contributions

The molecular descriptors are a useful tool in the spectral graph, molecular chemistry and several fields of chemistry and mathematics. The main contributions of this article are:

  • The edge F-index is proposed for fuzzy graphs here. Bounds of this index are calculated for FGs. The FG has been investigated for a given set of vertices as having maximum edge F-index. For an isomorphic FGs, it is shown that the value of this index is the same.

  • Some relations of this index with the second Zagreb index and hyper-Zagreb index are established.

  • Bounds of this index for some FG operations are determined.

  • An application of the index in mathematical chemistry is studied. For this, 18 octane isomers and 67 alkanes are considered and analyzed the correlation between this index with some properties of the octane isomers and alkanes. From the correlation coefficient value, we have obtained this index is highly correlated with enthalpy of vaporization, standard enthalpy of vaporization, entropy, acentric factor and heat of vaporization and less correlated with heat capacity for octane isomers. Also, this index is correlated with critical pressure, critical temperature, molar refraction, molar volume and boiling point and is less correlated with surface tension and heat of vaporization for alkanes. But, this index is inadequate to determine the melting point of alkanes.

Paper organization

The article’s structure: the next section provides some basic definitions. In the subsequent section, edge F-index is introduced and provides some bounds for FGs. For a given set of vertices, the maximal FGs are investigated with respect to this index here. Also, the value of this index is studied for isomorphic FGs here. In the penultimate section, bounds of this index for some FG operations are established. In the final section, a chemical applicability of the index is studied.

Preliminaries

Some useful definitions are given here, most of them are taken from [27, 28].

For a universal set X, a pair \(S = (X,\mu )\) is called a fuzzy set where \(\mu \) is a called membership function of S whose domain is X and co-domain is [0,1].

Definition 1

A FG is a triplet, \(G=(V,\sigma ,\mu )\), where V is called vertex set of the fuzzy graph with vertex membership function \(\sigma : V \rightarrow [0,1]\) and edge membership function \(\mu : V\times V \rightarrow [0,1]\) satisfying \(\mu (x,y) \le \min \{\sigma (x), \sigma (y)\}\).

The edge set of G is defined as \(E = \{(x,y)\in V\times V: \mu (x,y) > 0\}\). Note that, the edge (x, y) and (y, x) are considered as same and sometimes it called the edge xy or yx. Some times we have denoted \(G=(V,E)\) as a fuzzy graph with vertex set V and edge set E.

Degree of a vertex \(v \in V\) is defined as: \(d(v) = \sum _{x\in V}^{} \mu (xv).\) Let \(\varDelta \) and \(\delta \) be the maximum and minimum degree of G, respectively. Throughout this article, we consider, \( G_1 = (V_1, E_1)\) with \(n_1\)-vertices, \(m_1\)-edges and \( G_2 = (V_2, E_2)\) with \(n_2\)-vertices, \(m_2\)-edges be two FGs and \( \varDelta _1= \varDelta (G_1), \varDelta _2= \varDelta (G_2), \delta _1= \delta (G_1), \delta _2= \delta (G_2).\) Cartesian product, composition, join and union of two FGs is defined in [7]. Here direct product, strong product and semi-strong product of two FGs are defined.

Definition 2

Let \( G_1 = (V_1, E_1), G_2 = (V_2, E_2)\) be two FGs. Suppose, \(V_1 \cap V_2 = \phi \), then the direct product of \( G_1\) and \( G_2\) is a FG \( G_1\prod G_2 = (V, E)\), where \(V= V_1\times V_2\), \(E = \{((u_1,v_1)(u_2,v_2)): u_1u_2 \in E_1 \text { and } v_1v_2 \in E_2\}\), \(\sigma (u,v) = \sigma _1(u) \wedge \sigma _2(v)\) and \(\mu ((u_1,v_1)(u_2,v_2)) = \mu _1(u_1u_2) \wedge \mu _2 (v_1v_2)\).

Definition 3

Let \( G_1 = (V_1, E_1), G_2 = (V_2, E_2)\) be two FGs. Suppose, \(V_1 \cap V_2 = \phi \), then the semi-strong product of \( G_1\) and \( G_2\) is a FG \( G_1 \bullet G_2 = (V, E)\), where \(V= V_1\times V_2\), \(E = \{((u_1,v_1)(u_2,v_2)): u_1u_2 \in E_1 \text { and } v_1v_2 \in E_2\} \cup \{((u,v_1)(u,v_2)): u\in V_1, v_1v_2 \in E_2 \}\), \(\sigma (u,v) = \sigma _1(u) \wedge \sigma _2(v)\) and

$$\begin{aligned}{} & {} \mu ((u_1,v_1)(u_2,v_2)) \\{} & {} \quad ={\left\{ \begin{array}{ll} \mu _1(u_1u_2) \wedge \mu _2 (v_1v_2) &{}\text { if } u_1u_2\in E_1 \text { and } v_1v_2 \in E_2\\ \sigma _1(u) \wedge \mu _2 (v_1v_2) &{}\text { if } u_1 = u_2 = u \in V_1 \text { and } v_1v_2 \in E_2.\\ \end{array}\right. } \end{aligned}$$

Definition 4

Let \( G_1 = (V_1, E_1), G_2 = (V_2, E_2)\) be two FGs. Suppose, \(V_1 \cap V_2 = \phi \), then the strong product of \( G_1\) and \( G_2\) is a FG \( G_1 \otimes G_2 = (V, E)\), where \(V= V_1\times V_2\), \(E = \{((u_1,v_1)(u_2,v_2)): u_1u_2 \in E_1 \text { and } v_1v_2 \in E_2\} \cup \{((u,v_1)(u,v_2)): u\in V_1, v_1v_2 \in E_2 \}\), \(\sigma (u,v) = \sigma _1(u) \wedge \sigma _2(v)\) and

$$\begin{aligned}{} & {} \mu ((u_1,v_1)(u_2,v_2)) \\{} & {} \quad ={\left\{ \begin{array}{ll} \mu _1(u_1u_2) \wedge \mu _2 (v_1v_2) &{}\text { if } u_1u_2\in E_1 \text { and } v_1v_2 \in E_2\\ \sigma _1(u) \wedge \mu _2 (v_1v_2) &{}\text { if } u_1 = u_2 = u \in V_1 \text { and } v_1v_2 \in E_2\\ \sigma _2(v) \wedge \mu _1 (u_1u_2) &{}\text { if } u_1 u_2 \in E_1 \text { and } v_1 = v_2 = v \in V_2.\\ \end{array}\right. } \end{aligned}$$

The definition of ZI of crisp graph are

$$\begin{aligned}&{\textbf {First ZI: }} M_1(G) = \sum _{v\in V}^{} d^2(v) = \sum _{uv\in E}^{}[d(u) + d(v)].\\&{\textbf {Second ZI: }} M_2(G) = \sum _{uv\in E}^{} d(u)d(v). \end{aligned}$$

In [16], ZI for a FG is defined as

$$\begin{aligned}&{\textbf {First ZI: }} M(G) = \sum _{v\in V}^{} \sigma (v) d^2(v).\\&{\textbf {Second ZI: }} ZF_2(G) = \sum _{uv\in E}^{} \sigma (u) \sigma (v) d(u) d(v). \end{aligned}$$

Islam and Pal [7] modified the definition of first Zagreb index in FG as

$$\begin{aligned}&{\textbf {First ZI: }} ZF_1(G) = \sum _{v\in V}^{} [\sigma (v) d(v)]^2. \end{aligned}$$

Another topological indices for FGs are

$$\begin{aligned}&{\textbf {Edge first ZI: }} ZF_1^*(G) = \sum _{uv\in E}^{} [\sigma (u) d(u) + \sigma (v) d(v)].\\&{\textbf {Hyper-ZI: }} HZI(G) = \sum _{uv\in E}^{} [\sigma (u) d(u) + \sigma (v) d(v)]^2. \end{aligned}$$

Edge F-index for a fuzzy graph

In 2015, Fortula and Gutman introduced the F-index for crisp graph as:

$$\begin{aligned} F(G) = \sum _{u\in V}^{}d^3(u) = \sum _{uv\in E}[d^2(u) + d^2(v)]. \end{aligned}$$

For an edge \(uv\in E\), suppose, \(x = d(u) \ge d(v) = y\). The ordered pair (x, y) is called the degree coordinate of the edge uv. Then, \(x^2 + y^2\) is the square of the distance of the degree coordinate of uv from the origin. Hence, F-index is the sum of such square of the distances. Here the idea is introduced in the fuzzy graph.

Definition 5

For a FG G, the edge F-index for G is defined by

$$\begin{aligned} EFI(G) = \sum _{uv\in E}^{}\{[\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2\}. \end{aligned}$$

For an edge \(uv\in E\), we define \(x = \sigma (u) d(u) \ge \sigma (v) d(v) = y\). The order pair (x, y) is called the vertex-degree co-ordinate of the edge uv. Then \(x^2 + y^2\) is square of the distance of the vertex-degree co-ordinate of uv from origin. Hence, edge F-index is sum of such square of the distances.

Also, followed by the definition of the F-index for crisp graph, this index for a FG is introduced by Islam and Pal.

$$\begin{aligned} FI(G) = \sum _{u\in V}\sigma ^3(u) d^3(u). \end{aligned}$$

For crisp graph, the two definition of F-index are same but the next example shows that the two definition of F-index for a FG are different.

Fig. 1
figure 1

\(FI(G) \ne EFI(G)\)

Full size image

Example 1

Suppose G be a FG depicted in Fig. 1. Then, \(d(v_1) = 1.3, d(v_2) = 1.0, d(v_3) = 1.5, d(v_4) = 1.0\). Hence,

$$\begin{aligned}&FI(G) = \sum _{u\in V}\sigma ^3(u) d^3(u)\\&\quad \quad \quad \quad = [0.5\times 1.3]^2 + [0.6 \times 1.0]^2 + [0.7 \times 1.5]^2 \\&\quad \quad \quad \quad \quad + [0.6\times 1.0]^2\\&\quad \quad \quad \quad = 1.86425.\\&\text {But, } EFI(G) = \sum _{uv\in E}^{}[\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2 = 6.015. \end{aligned}$$

Therefore, for a FG G, \(FI(G) \ne EFI(G)\), in general.

Some bounds of this index are provided here.

Theorem 1

Suppose, G be a fuzzy graph. Then, \(EFI(G) \le 2\,m \varDelta ^2 \le 2\,m (n-1)^2\).

Now, this index is studied for a partial fuzzy subgraph (Partial-FSG).

Theorem 2

Suppose H be a partial-FSG of a FG G. Then, \(EFI(H) \le EFI(G).\)

Proof

As H be a partial-FSG of G, then, \(0 \le \sigma _H(v) \le \sigma _G(v), \mu _H(uv)\le \mu _G(uv), \forall v\in V(H) \subset V(G), \forall uv\in E(H) \subset E(G).\) Therefore, for \(v\in V(H)\),

$$\begin{aligned} d_H (v)&= \sum _{u\in V(H)}^{} \mu _H(uv) \le \sum _{u\in V(G)}^{}\mu _G(uv) = d_G (v).\\ \text { Hence, } EFI(H)&= \sum _{uv\in E(H)}[\sigma _H (u) d_H(u)]^2 + [\sigma _H (v) d_H(v)]^2\\&\le \sum _{uv\in E(G)}[\sigma _G (u) d_G(u)]^2 + [\sigma _G (v) d_G(v)]^2\\&= FHZI(G). \end{aligned}$$

As a FSG is also a partial-FSG, the above result is also true for a FSG.

Corollary 1

Suppose, V be a given set of vertices. Then complete-FG spanned by V has maximum edge F-index among the FGs spanned by V.

The edge F-index for isomorphic FGs are studied below.

Theorem 3

Suppose \(G_1\) and \(G_2\) be two isomorphic FGs. Then, \(EFI(G_1) = EFI(G_2)\).

Proof

As \(G_1\) and \(G_2\) be two isomorphic FGs, there exist a bijective map \(\psi : V(G_1) \rightarrow V(G_2)\) such that, \(\sigma _{G_1}(v) = \sigma _{G_2}(\psi (v))\) and \(\mu _{G_1}(uv) = \mu _{G_2}(\psi (u)\psi (v))\), \(\forall u,v\in V(G_1) \) and \(u\not = v.\) Then

$$\begin{aligned} d_{G_1}(v)&= \sum _{u \in V{G_1}}^{}\mu _{G_1}(uv) \\&\quad = \sum _{\psi (u) \in V{G_2}}^{} \mu _{G_2}(\psi (u)\psi (v)) = d_{G_2}(\psi (v)). \end{aligned}$$
$$\begin{aligned}&\text { Therefore, } EFI(G_1) \\&\quad = \sum _{uv\in E(G_1)}[\sigma _{G_1} (u) d_{G_1}(u)]^2 \\&\qquad + [\sigma _{G_1} (v) d_{G_1}(v)]^2\\&\quad = \sum _{\psi (u)\psi (v)\in E(G_2)}[\sigma _{G_2} (\psi (u)) d_{G_2}(\psi (u))]^2 \\&\qquad + [\sigma _{G_2} (\psi (v)) d_{G_2}(\psi (v))]^2\\&\quad = EFI(G_2). \end{aligned}$$

Relation of edge F-index with other topological indices for fuzzy graph

Here, we have established some relations of this index with other TIs for fuzzy graph. First we have studied the relation between this index and second Zagreb index for fuzzy graphs.

Theorem 4

Suppose G be a FG. Then, \(EFI (G) \ge 2 ZF_2 (G)\).

Proof

For any two real numbers x and y,

$$\begin{aligned}&(x-y)^2 \ge 0\nonumber \\ \text { or }&x^2 + y^2 \ge 2 xy. \end{aligned}$$
(1)

In Eq. (1), put \(x = \sigma (u) d(u)\) and \(y = \sigma (v)d(v)\), we get,

$$\begin{aligned}&[\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2 \ge 2 \sigma (u) d(u) \sigma (v) d(v)\\ \text { or }&\sum _{uv\in E}^{}[\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2\\&\quad \ge 2 \sum _{uv\in E}^{}\sigma (u) d(u) \sigma (v) d(v)\\ \text { or }&EFI(G) \ge ZF_2(G). \end{aligned}$$

The next theorem provides the relation this index and hyper-Zagreb index for FGs.

Theorem 5

Suppose G be a FG. Then, \(EFI (G) \ge \frac{1}{2} HZI (G)\).

Proof

For any two real numbers x and y, from Eq. (1), we have

$$\begin{aligned}&(x^2 + y^2) \ge \frac{1}{2} (x+y)^2. \end{aligned}$$
(2)

In Eq. (2), put \(x = \sigma (u) d(u)\) and \(y = \sigma (v)d(v)\), we get

$$\begin{aligned}&[\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2 \ge \frac{1}{2} [\sigma (u) d(u) + \sigma (v) d(v)]^2\\&\text { or } \sum _{uv\in E}^{}[\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2 \\&\qquad \quad \ge \frac{1}{2} \sum _{uv\in E}^{}[\sigma (u) d(u) + \sigma (v) d(v)]^2\\&\text { or } EFI(G) \ge \frac{1}{2} HZI (G). \end{aligned}$$

The relation among edge F-index, hyper-ZI and second ZI for fuzzy graphs is determined below.

Theorem 6

Suppose G be a FG. Then, \(EFI(G) = HZI (G) - 2ZF_2(G)\).

Proof

For \(u,v \in V\), we have,

$$\begin{aligned}&[\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2\\&\quad = [\sigma (u)d(u) + \sigma (v)d(v)]^2 - 2\sigma (u)d(u)\sigma (v)d(v)\\&\text { or } \sum _{uv\in E}^{}[\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2 \\&= \sum _{uv\in E}^{}[\sigma (u)d(u) + \sigma (v)d(v)]^2\\&\qquad - 2 \sum _{uv\in E}^{} \sigma (u)d(u)\sigma (v)d(v)\\&\text { or } EFI(G) = HZI(G) - 2ZF_2(G). \end{aligned}$$

Edge F-index for fuzzy graph operations

In this section, edge F-index is studied for some FG operations.

Theorem 7

\(EFI(G_1 \times G_2) \le 2[m_1ZF_1(G_2) + m_2ZF_1(G_1)] + 2[n_1\varDelta _1ZF_1^*(G_2) + n_2\varDelta _2 ZF_1^*(G_1)] + [n_1EFI(G_2) + n_2EFI(G_1)]\).

Proof

As \( G_1\times G_2\) is Cartesian product of \( G_1\) and \( G_2\), then for \((u,v), (u_1,v_1) \in V, \sigma (u,v) = \wedge \{\sigma _1(u), \sigma _2(v)\}\) and

$$\begin{aligned}{} & {} \mu ((u,v)(u_1, v_1)) \\{} & {} \quad = {\left\{ \begin{array}{ll} \wedge \{\sigma _1(u), \mu _2(v,v_1)\} \text { if } u = u_1 \text { and } v v_1\in E_2\\ \wedge \{\sigma _2(v), \mu _1(u,u_1)\} \text { if } u u_1\in E_1 \text { and } v = v_1\\ 0 \text { otherwise.} \end{array}\right. } \end{aligned}$$
$$\begin{aligned} \text { Then, } d(u,v)&= \sum _{v_i v\in E_2}^{} \mu \{(u,v),(u,v_i)\} \\&\quad + \sum _{u_i u\in E_1}^{} \mu \{(u,v),(u_i,v)\}\\&= \sum _{v_i v\in E_2}^{} \wedge \{\sigma _1(u) , \mu _2(v,v_i)\} \\&\quad + \sum _{u_i u \in E_1}^{} \wedge \{\sigma _2(v) , \mu _1(u,u_i)\}\\&\le \sum _{v_i v\in E_2}^{} \mu _2(v,v_i) + \sum _{u_i u\in E_1}^{} \mu _1(u,u_i)\\&= d_1(u) + d_2(v). \end{aligned}$$

Then the edge F-index of \( G_1 \times G_2\) is

$$\begin{aligned}&EFI(G_1\times G_2) \\&\quad = \sum _{(u_1,v_1)(u_2,v_2) \in E} [\sigma (u_1,v_1) d(u_1,v_1)]^2 \\&\qquad + [\sigma (u_2,v_2) d(u_2,v_2)]^2\\&\quad = \underbrace{\sum _{u\in V_1, v_1v_2 \in E_2} [\sigma (u,v_1) d(u,v_1)]^2 + [\sigma (u,v_2) d(u,v_2)]^2}_{K_1} \\&\qquad + \underbrace{\sum _{v \in V_2, u_1u_2 \in E_1} [\sigma (u_1,v) d(u_1,v)]^2 + [\sigma (u_2,v) d(u_2,v)]^2}_{K_2}.\\&\text { Now, } K_1\\&\quad = \sum _{u\in V_1, v_1v_2 \in E_2} [\sigma (u,v_1) d(u,v_1)]^2 + [\sigma (u,v_2) d(u,v_2)]^2\\&\quad \le \sum _{u\in V_1, v_1v_2 \in E_2} \left[ \{\sigma _1(u) d_1(u) + \sigma _2(v_1) d_2(v_1)\}^2 \right. \\&\qquad \left. + \{\sigma _1(u) d_1(u) + \sigma _2(v_2) d_2(v_2)\}^2 \right] \\&\quad = \sum _{u\in V_1, v_1v_2 \in E_2} \Big [2 \sigma _1^2(u)d_1^2(u) + 2 \sigma _1(u)d_1(u)\{\sigma _2(v_1)d_2(v_1) \\&\qquad + \sigma _2(v_2)d_2(v_2)\} + \{\sigma ^2_2(v_1)d^2_2(v_1) + \sigma ^2_2(v_2)d^2_2(v_2)\} \Big ]\\&\therefore K_1 \le 2m_2ZF_1(G_1) + 2n_1\varDelta _1 ZF_1^*(G_2) + n_1 EFI(G_2)\\&\text { Similarly, } K_2 \\&\quad \le 2m_1ZF_1(G_2) + 2n_2\varDelta _2 ZF_1^*(G_1) + n_2 EFI(G_1). \end{aligned}$$

Hence, the result follows.

Corollary 2

\(EFI(G_1 \times G_2) \le (n_1m_2 + n_2m_1) (\varDelta _1 + \varDelta _2)^2 \).

Proof

By Theorem 7, we have

$$\begin{aligned} EFI(G_1 \times G_2)&\le 2[m_1ZF_1(G_2) + m_2ZF_1(G_1)] \\&+ 2[n_1\varDelta _1ZF_1^*(G_2) + n_2\varDelta _2 ZF_1^*(G_1)]\\&+ [n_1EFI(G_2) + n_2EFI(G_1)]. \end{aligned}$$

Also, the following are holds:

$$\begin{aligned}&ZF_1(G) \le n \varDelta ^2\\&ZF_1^*(G) \le 2m \varDelta \\&\text { and } EFI(G) \le 2m \varDelta ^2.\\ \end{aligned}$$

From those results and Theorem 7, the result follows.

Theorem 8

\( EFI(G_1[G_2]) \le n_2^4EFI(G_1) + 2n_2^2m_2ZF_1 (G_1) + 4n_2^3\varDelta _2 ZF_1^*(G_1) + 2n_1n_2\varDelta _1ZF_1^*(G_2) + n_1EFI(G_2) + 2m_1ZI(G_2).\)

Proof

As \( G_1[G_2]\) is composition graph of \( G_1\) and \( G_2\), then for \((u,v), (u_1,v_1) \in V, \sigma (u,v) = \wedge \{\sigma _1(u), \sigma _2(v)\}\) and

$$\begin{aligned}{} & {} \mu ((u,v)(u_1, v_1)) \nonumber \\{} & {} \quad = {\left\{ \begin{array}{ll} \wedge \{\sigma _1(u), \mu _2(v,v_1)\} &{}\text { if } u= u_1 \text { and } v v_1\in E_2\\ \wedge \{\sigma _2(v), \sigma _2(v_1), \mu _1(u,u_1)\} &{}\text { if } u u_1\in E_1\\ 0 &{}\text { otherwise.} \end{array}\right. } \end{aligned}$$

Then,

$$\begin{aligned} d(u,v)&= \sum _{v_i v\in E_2}^{} \mu \{(u,v),(u,v_i)\}\\&\quad + \sum _{u_i u\in E_1, v_j\in V_2}^{} \mu \{(u,v),(u_i,v_j)\}\\&= \sum _{v_i v\in E_2}^{} \wedge \{\sigma _1(u) , \mu _2(v,v_i)\} \\&\quad + \sum _{u_i u\in E_1, v_j \in V_2}^{} \wedge \{\sigma _2(v),\sigma _2(v_j) \mu _1(u,u_i)\}\\&\le \sum _{v_i v\in E_2}^{} \mu _2(v,v_i) + \sum _{u_i u\in E_1, v_j \in V_2}^{} \mu _1(u,u_i)\\&= n_2 d_1(u) + d_2(v). \end{aligned}$$

Then the edge index of \(G_1[G_2]\) is

$$\begin{aligned}&EFI(G_1[G_2]) \\&\quad = \sum _{(u_1,v_1)(u_2,v_2) \in E} [\sigma (u_1,v_1) d(u_1,v_1)]^2 + [\sigma (u_2,v_2) d(u_2,v_2)]^2\\&\quad = \underbrace{\sum _{u\in V_1, v_1v_2 \in E_2} [\sigma (u,v_1) d(u,v_1)]^2 + [\sigma (u,v_2) d(u,v_2)]^2}_{K_1} \\&\qquad + \underbrace{\sum _{u_1u_2 \in E_1, v_1,v_2 \in V_2} [\sigma (u_1,v_1) d(u_1,v_1)]^2 + [\sigma (u_2,v_2) d(u_2,v_2)]^2}_{K_2}. \\&\text { Now, } K_1 \\&\quad = \sum _{u\in V_1, v_1v_2 \in E_2} [\sigma (u,v_1) d(u,v_1)]^2 + [\sigma (u,v_2) d(u,v_2)]^2\\&\quad \le \sum _{u\in V_1, v_1v_2 \in E_2} [n_2\sigma _1(u) d_1(u) + \sigma _2(v_1) d_2(v_1)]^2 \\&\qquad + [n_2\sigma _1(u) d_1(u) + \sigma _2(v_2) d_2(v_2)]^2\\&\quad = \sum _{u\in V_1, v_1v_2 \in E_2} \left[ 2n_2^2 \sigma _1^2(u) d_1^2(u) \right. \\&\qquad \left. + 2n_2\sigma _1(u)d_1(u)\{ \sigma _2(v_1)d_2(v_1) +\sigma _2(v_2)d_2(v_2) \}\right. \\&\qquad \left. + \{ \sigma _2^2(v_1)d_2^2(v_1) + \sigma _2^2(v_2)d_2^2(v_2) \}\right] \\&\quad \le 2n_2^2m_2ZF_1(G_1) + 2n_1n_2\varDelta _1ZF_1^*(G_2) + n_1 EFI(G_2).\\&\text { Again, } K_2\\&\quad = \sum _{u_1u_2 \in E_1,v_1,v_2 \in V_2} [\sigma (u_1,v_1) d(u_1,v_1)]^2 \\&\qquad + [\sigma (u_2,v_2) d(u_2,v_2)]^2\\&\quad \le \sum _{u_1u_2 \in E_1, v_1,v_2 \in V_2} [n_2\sigma _1(u_1) d_1(u_1) + \sigma _2(v_1) d_2(v_1)]^2 \\&\qquad + [n_2\sigma _1(u_2) d_1(u_2) + \sigma _2(v_2) d_2(v_2)]^2\\&\quad \le \sum _{u_1u_2 \in E_1, v_1,v_2 \in V_2}\left[ n_2^2 \{\sigma _1^2(u_1) d_1^2(u_1) + \sigma _1^2(u_2) d_1^2(u_2)\}\right. \\&\qquad \left. + \{ \sigma _2^2(v_1)d_2^2(v_1) + \sigma _2^2(v_2)d_2^2(v_2) \}\right. \\&\qquad \left. + 2n_2\{ \sigma _1(u_1)d_1(u_1) + \sigma _1(u_2)d_1(u_2)\}\right. \\&\qquad \left. \{ \sigma _2(v_1)d_2(v_1) + \sigma _2(v_2)d_2(v_2) \} \right] \\&\quad \le n^4_2 EFI(G_1) + 2m_1 ZF_1(G_2) + 4n_2^3\varDelta _2 ZF_1^*(G_1). \end{aligned}$$

Hence, the result follows.

Theorem 9

\(EFI(G_1 + G_2) \ge HZI(G_1) + HZI(G_2) + n_2ZF_1(G_1) + n_1ZF_1(G_2) -2ZF_2(G_1) - 2ZF_2(G_2)\).

Proof

As \( G_1+ G_2\) is join graph of \( G_1\) and \( G_2\), then for \(u,v,u_1,v_1 \in V\),

$$\begin{aligned} \sigma (u) = {\left\{ \begin{array}{ll} \sigma _1(u) \text { if } u\in V_1\\ \sigma _2(u) \text { if } u\in V_2 \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \mu (u,v) = {\left\{ \begin{array}{ll} \wedge \{\sigma _1(u), \sigma _2(v)\} \text { if } u\in V_1 \text { and } v\in V_2\\ \mu _1(u,v) \text { if } uv\in E_1\\ \mu _2(u,v) \text { if } uv\in E_2\\ 0 \text { otherwise.} \end{array}\right. } \end{aligned}$$

Then,

$$\begin{aligned} d(u)&= {\left\{ \begin{array}{ll} d_1(u) + \sum _{v\in V_2}^{} \wedge \{\sigma _1(u), \sigma _2(v)\} \text { if } u\in V_1\\ d_2(u) + \sum _{v\in V_1}^{} \wedge \{\sigma _2(u), \sigma _1(v)\} \text { if } u\in V_2 \end{array}\right. }\\&\ge {\left\{ \begin{array}{ll} d_1(u) \text { if } u\in V_1\\ d_2(u) \text { if } u\in V_2. \end{array}\right. } \end{aligned}$$

Then the edge F-index of \( G_1+ G_2\) is

$$\begin{aligned}&EFI(G_1+G_2) = \sum _{uv\in E} [\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2\\&= \underbrace{\sum _{uv\in E_1} [\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2}_{K_1} \\&\quad + \underbrace{\sum _{uv\in E_2} [\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2}_{K_2} \\&+ \underbrace{\sum _{u\in V_1, v\in V_2} [\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2}_{K_3}\\ \\&\text { Now, } K_1 = \sum _{uv\in E_1} [\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2\\&\ge \sum _{uv\in E_1} [\sigma _1(u)d_1(u)]^2 + [\sigma _1(v)d_1(v)]^2\\&= \sum _{uv\in E_1} [\sigma _1(u)d_1(u) + \sigma _1(v)d_1(v)]^2 \\&\quad -2\sum _{uv\in E_1} \sigma _1(u)d_1(u) \sigma _1(v)d_1(v) \\ \therefore K_1&\ge HZI(G_1) -2ZF_2(G_1).\\&\text { Similarly, } K_2 \ge HZI(G_2) -2ZF_2(G_2).\\&\text { Again, } K_3 = \sum _{u\in V_1, v\in V_2} [\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2\\&\ge \sum _{u\in V_1, v\in V_2} [\sigma _1(u)d_1(u)]^2 \\&\quad + \sum _{u\in V_1, v\in V_2} [\sigma _2(v)d_2(v)]^2\\&\therefore K_3 \ge n_2ZF_1(G_1) \\&\quad + n_1ZF_1(G_2). \end{aligned}$$

Hence, the result follows.

Theorem 10

\(EFI(G_1 + G_2) \le EFI(G_1) + EFI(G_2) + 2n_2ZF_1(G_1) + 2n_1ZF_1(G_2) +n_2ZF_1(G_1) + n_1ZF_1(G_2) + 2[n_1^2m_2 + n_2^2m_1] + n_1n_2[n_1^2 + n_2^2] + 2n_1n_2[n_2\varDelta _1 + n_1\varDelta _2]\).

Proof

As \( G_1+ G_2\) is join graph of \( G_1\) and \( G_2\), then for \(u,v,u_1,v_1 \in V\),

$$\begin{aligned} \sigma (u) = {\left\{ \begin{array}{ll} \sigma _1(u) \text { if } u\in V_1\\ \sigma _2(u) \text { if } u\in V_2 \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \mu (u,v) = {\left\{ \begin{array}{ll} \wedge \{\sigma _1(u), \sigma _2(v)\} \text { if } u\in V_1 \text { and } v\in V_2\\ \mu _1(u,v) \text { if } uv\in E_1\\ \mu _2(u,v) \text { if } uv\in E_2\\ 0 \text { otherwise.} \end{array}\right. } \end{aligned}$$

Then,

$$\begin{aligned} d(u)&= {\left\{ \begin{array}{ll} d_1(u) + \sum _{v\in V_2}^{} \wedge \{\sigma _1(u), \sigma _2(v)\} \text { if } u\in V_1\\ d_2(u) + \sum _{v\in V_1}^{} \wedge \{\sigma _2(u), \sigma _1(v)\} \text { if } u\in V_2 \end{array}\right. }\\&\le {\left\{ \begin{array}{ll} d_1(u) + \sum _{v\in V_2}^{}\sigma _1(u), \text { if } u\in V_1\\ d_2(u) + \sum _{v\in V_1}^{}\sigma _2(u), \text { if } u\in V_2 \end{array}\right. }\\&\le {\left\{ \begin{array}{ll} d_1(u) + n_2 \text { if } u\in V_1\\ d_2(u) + n_1 \text { if } u\in V_2. \end{array}\right. } \end{aligned}$$

Then the edge F-index of \( G_1+ G_2\) is:

$$\begin{aligned} EFI(G_1+G_2)&= \sum _{uv\in E} [\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2\\&= \underbrace{\sum _{uv\in E_1} [\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2}_{K_1} \\&\quad + \underbrace{\sum _{uv\in E_2} [\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2}_{K_2} \\&+ \underbrace{\sum _{u\in V_1, v\in V_2} [\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2}_{K_3}\\\\ \text { Now, } K_1&= \sum _{uv\in E_1} [\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2\\&\le \sum _{uv\in E_1} [\sigma _1(u)d_1(u) + n_2]^2 + [\sigma _1(v)d_1(v) + n_2]^2\\&= \sum _{uv\in E_1} [\sigma _1^2(u)d_1^2(u) + \sigma _1^2(v)d_1^2(v)] \\&\quad + 2n_2 \sum _{uv\in E_1} [\sigma _1(u)d_1(u) + \sigma _1(v)d_1(v)] + 2n_2^2 \\ \therefore K_1&\le EFI(G_1) + 2n_2 ZF_1^*(G_1) + 2n_2^2m_1.\\ \text { Similarly, } K_2&\le EFI(G_2) + 2n_1 ZF_1^*(G_2) + 2n_1^2m_2.\\ \text { Again, } K_3&= \sum _{u\in V_1, v\in V_2} [\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2\\&\le \underbrace{\sum _{u\in V_1, v\in V_2} [\sigma _1(u)d_1(u) + n_2]^2}_{P_1} \\&\quad + \underbrace{\sum _{u\in V_1, v\in V_2} [\sigma _2(v)d_2(v) + n_1]^2}_{P_2}\\ \text { Now, } P_1&= \sum _{u\in V_1, v\in V_2} [\sigma _1(u)d_1(u) + n_2]^2\\&= \sum _{u\in V_1, v\in V_2} [\sigma _1^2(u)d^2_1(u) + 2n_2 \sigma _1(u)d_1(u) + n_2^2]\\ \therefore P_1&\le n_2ZF_1(G_1) + 2n_1n_2^2\varDelta _1 +n_1n_2^3. \\ \text { Similarly, } P_2&\le n_1ZF_1(G_2) + 2n_2n_1^2\varDelta _1 +n_2n_1^3.\\ \text { Therefore, } K_3&\le n_2ZF_1(G_1) + n_1ZF_1(G_2)\\&\quad + n_1n_2[n_1^2 + n_2^2] + 2n_1n_2[n_2\varDelta _1 + n_1\varDelta _2]. \end{aligned}$$

Hence, the result follows.

Theorem 11

\( EFI( G_1\cup G_2) \ge EFI( G_1) + EFI( G_2) -2p\varDelta ^2\), where \(p = |E_1\cap E_2|, \varDelta = \max \{\varDelta _1,\varDelta _2\}\).

Proof

As \(G_1 \cup G_2\) is union of \(G_1\) and \(G_2\), then for any \(u,v\in V,\)

$$\begin{aligned}&\sigma (u) = {\left\{ \begin{array}{ll} \sigma _1(u) \text { if } u\in V_1\setminus V_2\\ \sigma _2(u) \text { if } u\in V_2\setminus V_1\\ \vee \{\sigma _1(u),\sigma _2(u)\} \text { if } u\in V_1\cap V_2 \end{array}\right. }\\&\text { and } \mu (uv) = {\left\{ \begin{array}{ll} \mu _1(uv) \text { if } uv\in E_1\setminus E_2\\ \mu _2(uv), \text { if } uv\in E_2\setminus E_1\\ \vee \{\mu _1(uv), \mu _2(uv)\}, \text { if } uv\in E_1\cap E_2\\ 0 \text { otherwise.} \end{array}\right. }\\&\text { Then, } d(u) \ge {\left\{ \begin{array}{ll} d_1(u), \text { if } u\in V_1\setminus V_2\\ d_2(u), \text { if } u\in V_2\setminus V_1\\ d_1(u) \text { or } d_2(u), \text { if } u\in V_1\cap V_2. \end{array}\right. } \end{aligned}$$

Hence, the edge F-index for \(G_1\cup G_2\) is:

$$\begin{aligned} EFI(G_1\cup G_2)&= \sum _{uv\in E}^{}[\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2\\&= \sum _{uv\in E_1} [\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2 \\&\quad + \sum _{uv\in E_2}^{}[\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2 \\&\quad - \sum _{uv\in E_1\cap E_2}^{}[\sigma (u)d(u)]^2 + [\sigma (v)d(v)]^2 \\&\ge EFI(G_1) + EFI(G_2) - 2p\varDelta ^2. \end{aligned}$$

Theorem 12

\(EFI(G_1 \otimes G_2) \le 8 [EFI(G_1) + EFI(G_2) + m_2ZF_1(G_1) + m_1ZF_1(G_2) + ZF_1^*(G_1)ZF_1^*(G_2) + n_1\varDelta _1 ZF_1^*(G_1) + n_2\varDelta _2ZF_1^*(G_2)]\).

Proof

As \(G_1 \otimes G_2\) is strong product graph of \(G_1\) and \(G_2\), then, \(\sigma (u,v) = \sigma _1(u) \wedge \sigma _2(v)\) and

$$\begin{aligned}&\mu ((u_1,v_1)(u_2,v_2)) \\&\quad = {\left\{ \begin{array}{ll} \mu _1(u_1u_2) \wedge \mu _2 (v_1v_2) \text { if } u_1u_2\in E_1 \text { and } v_1v_2 \in E_2\\ \sigma _1(u) \wedge \mu _2 (v_1v_2) \text { if } u_1 = u_2 = u \in V_1 \text { and } v_1v_2 \in E_2\\ \sigma _2(v) \wedge \mu _1 (u_1u_2) \text { if } u_1 u_2 \in E_1 \text { and } v_1 = v_2 = v \in V_2.\\ \end{array}\right. } \end{aligned}$$

Hence,

$$\begin{aligned} d (u,v) =&\sum _{uu_1 \in E_1, vv_1 \in E_2}^{} \mu ((u,v)(u_1,v_1))\\&+ \sum _{vv_1 \in E_2}^{} \mu ((u,v)(u,v_1))\\&+ \sum _{uu_1 \in E_1}^{} \mu ((u,v)(u_1,v))\\ =&\sum _{uu_1 \in E_1, vv_1 \in E_2}^{} \mu _1(uu_1) \wedge \mu _2 (vv_1)\\&+ \sum _{vv_1 \in E_2}^{} \sigma _1(u) \wedge \mu _2 (vv_1)\\&+ \sum _{uu_1 \in E_1}^{} \sigma _2(v) \wedge \mu _1 (uu_1) \\ \le&\sum _{uu_1 \in E_1}^{} \mu _1(uu_1) + \sum _{vv_1 \in E_2}^{} \mu _2 (vv_1)\\&+ \sum _{vv_1 \in E_2}^{} \mu _2 (vv_1) + \sum _{uu_1 \in E_1}^{} \mu _1 (uu_1)\\ =&2[d_1(u) + d_2(v)]. \end{aligned}$$

Therefore, the edge F-index of \(G_1 \otimes G_2\) is:

$$\begin{aligned}&EFI(G_1 \otimes G_2) = \sum _{(u_1,v_1)(u_2,v_2)\in E}^{} [\sigma (u_1,v_1)d(u_1,v_1)]^2 \\&\quad + [\sigma (u_2,v_2)d(u_2,v_2)]^2\\&= \underbrace{\sum _{u_1u_2 \in E_1, v_1v_2 \in E_2}^{} [\sigma (u_1,v_1)d(u_1,v_1)]^2 + [\sigma (u_2,v_2)d(u_2,v_2)]^2}_{K_1} \\&\quad + \underbrace{\sum _{u \in V_1, v_1v_2 \in E_2}^{} [\sigma (u,v_1)d(u,v_1)]^2 + [\sigma (u,v_2)d(u,v_2)]^2}_{K_2} \\&\quad + \underbrace{\sum _{u_1u_2 \in E_1, v \in V_2}^{}[\sigma (u_1,v)d(u_1,v)]^2 + [\sigma (u_2,v)d(u_2,v)]^2}_{K_3}.\\&\text { Now, } K_1 = \sum _{u_1u_2 \in E_1, v_1v_2 \in E_2}^{} [\sigma (u_1,v_1)d(u_1,v_1)]^2 \\&\quad + [\sigma (u_2,v_2)d(u_2,v_2)]^2\\&\le 4 \sum _{u_1u_2 \in E_1, v_1v_2 \in E_2}^{} [\sigma _1(u_1)d_1(u_1) + \sigma _2(v_1)d_2(v_1)]^2 \\&\quad + [\sigma _1(u_2)d_1(u_2) + \sigma _2(v_2)d_2(v_2)]^2\\&\le 4 \sum _{u_1u_2 \in E_1, v_1v_2 \in E_2}^{} [\{\sigma _1^2(u_1)d_1^2(u_1) + \sigma _1^2(u_2)d_1^2(u_2) \} \\&\quad + \{\sigma _2^2(v_1)d_2^2(v_1) + \sigma _2^2(v_2)d_2^2(v_2) \} \\&+ 2\{ \sigma _1(u_1)d_1(u_1) + \sigma _1(u_2)d_1(u_2)\} \{\sigma _2(v_1)d_2(v_1) \\&\quad + \sigma _2(v_2)d_2(v_2)\}].\\&\therefore K_1 \le 4[EFI(G_1) + EFI(G_2) \\&\quad + 2ZF_1^*(G_1)ZF_1^*(G_2)].\\&\text { Again, } K_2 = \sum _{u \in V_1, v_1v_2 \in E_2}^{} [\sigma (u,v_1)d(u,v_1)]^2 \\&\quad + [\sigma (u,v_2)d(u,v_2)]^2\\&\le 4 \sum _{u \in V_1, v_1v_2 \in E_2}^{} [\sigma _1(u)d_1(u) + \sigma _2(v_1)d_2(v_1)]^2 \\&\quad + [\sigma _1(u)d_1(u) + \sigma _2(v_2)d_2(v_2)]^2\\&= 4 \sum _{u \in V_1, v_1v_2 \in E_2}^{} [2\sigma _1^2(u)d_1^2(u) \\&\quad + 2\sigma _1(u)d_1(u)\{ \sigma _2(v_1)d_2(v_1) + \sigma _2(v_2)d_2(v_2) \} \\&\quad + \{ \sigma _2^2(v_1)d_2^2(v_1) + \sigma _2^2(v_2)d_2^2(v_2) \}]\\&\therefore K_2 \le 4 [2m_2ZF_1(G_1) + 2n_1\varDelta _1ZF_1^*(G_2) + EFI(G_2)].\\&\text { Similarly, } K_3 \le 4 [2m_1ZF_1(G_2) + n_2\varDelta _2ZF_1^*(G_1)\\&\quad + EFI(G_1)]. \end{aligned}$$

Hence, the result follows.

Theorem 13

\(EFI(G_1 \prod G_2) \le \frac{1}{4} [m_2EFI(G_1) + m_1 EFI(G_2) + 2ZF_1^*(G_1)ZF_1^*(G_2)]\).

Proof

As \(G_1 \prod G_2\) is direct product graph of \(G_1\) and \(G_2\), then, \(\sigma (u,v) = \sigma _1(u) \wedge \sigma _2(v)\) and \(\mu ((u_1,v_1)(u_2,v_2)) = \mu _1(u_1u_2) \wedge \mu _2 (v_1v_2)\). Hence,

$$\begin{aligned} d (u,v)&= \sum _{(u_1,v_1) \in V}^{} \mu ((u,v)(u_1,v_1))\\&= \sum _{(u_1,v_1) \in V}^{} \mu _1(u,u_1) \wedge \mu _2(v,v_1)\\&\le d_1(u) \wedge d_2(v)\\&\le \frac{1}{2} [d_1(u) + d_2(v)]. \end{aligned}$$

Therefore, the edge F-index for \(G_1 \prod G_2\) is:

$$\begin{aligned}&EFI(G_1 \prod G_2) = \sum _{(u_1,v_1)(u_2,v_2)\in E}^{} [\sigma (u_1,v_1)d(u_1,v_1)]^2 \\&\quad + [\sigma (u_2,v_2)d(u_2,v_2)]^2\\&\le \frac{1}{4} \sum _{u_1u_2 \in E_1, v_1v_2 \in E_2}^{} [\sigma _1(u_1)d_1(u_1) + \sigma _2(v_1)d_2(v_1)]^2 \\&\quad + [\sigma _1(u_2)d_1(u_2) + \sigma _2(v_2)d_2(v_2)]^2\\&\le \frac{1}{4} \sum _{(u_1,v_1)(u_2,v_2)\in E}^{} [\{\sigma _1^2(u_1)d_1^2(u_1) + \sigma _1^2(u_2)d_1^2(u_2)\} \\&\quad + \{\sigma _2^2(v_1)d_2^2(v_1) + \sigma _2^2(v_2)d_2^2(v_2)\} \\&\quad + 2 \{\sigma _1(u_1)d_1(u_1) + \sigma _1(u_2)d_1(u_2)\}\\&\quad \{\sigma _2(v_1)d_2(v_1) + \sigma _2(v_2)d_2(v_2)\}]\\&\le \frac{1}{4} [m_2EFI(G_1) + m_1EFI(G_2) + 2ZF_1^*(G_1)ZF_1^*(G_2)]. \end{aligned}$$

Theorem 14

\(EFI(G_1 \bullet G_2) \le 2m_2ZF_1(G_1) + m_2EFI(G_1) + 4(m_1+n_1) EFI(G_2) + 4n_1\varDelta _1ZF_1^*(G_2) + 4ZF_1^*(G_1) ZF_1^*(G_2)\).

Proof

As \(G_1 \bullet G_2\) is semi-strong product graph of \(G_1\) and \(G_2\), then, \(\sigma (u,v) = \sigma _1(u) \wedge \sigma _2(v)\) and

$$\begin{aligned}&\mu ((u_1,v_1)(u_2,v_2)) \\&\quad = {\left\{ \begin{array}{ll} \mu _1(u_1u_2) \wedge \mu _2 (v_1v_2) &{}\text { if } u_1u_2 \in E_1 \text { and } v_1v_2 \in E_2\\ \sigma _1(u) \wedge \mu _2 (v_1v_2) &{}\text { if } u_1 = u_2 = u \in V_1 \text { and }\\ &{} v_1v_2 \in E_2. \end{array}\right. } \end{aligned}$$

Hence,

$$\begin{aligned} d (u,v)&= \sum _{uu_1 \in E_1, vv_1 \in E_2}^{} \mu ((u,v)(u_1,v_1)) \\&\quad + \sum _{vv_1 \in E_2}^{} \mu ((u,v)(u,v_1)) \\&= \sum _{uu_1 \in E_1, vv_1 \in E_2}^{} \mu _1(uu_1) \wedge \mu _2 (vv_1) \\&\quad + \sum _{vv_1 \in E_2}^{} \sigma _1(u) \wedge \mu _2 (vv_1) \\&\le \sum _{uu_1 \in E_1}^{} \mu _1(uu_1) + \sum _{vv_1 \in E_2}^{} \mu _2 (vv_1) \\&\quad + \sum _{vv_1 \in E_2}^{} \mu _2 (vv_1)\\&= d_1(u) + 2d_2(v). \end{aligned}$$

Therefore, the edge F-index for \(G_1 \bullet G_2\) is:

$$\begin{aligned}&EFI(G_1 \bullet G_2)\\&\quad = \sum _{(u_1,v_1)(u_2,v_2)\in E}^{} [\sigma (u_1,v_1)d(u_1,v_1)]^2 \\&\qquad + [\sigma (u_2,v_2)d(u_2,v_2)]^2\\&\quad = \underbrace{\sum _{u_1u_2\in E_1, v_1v_2\in E_2}^{} [\sigma (u_1,v_1)d(u_1,v_1)]^2 + [\sigma (u_2,v_2)d(u_2,v_2)]^2}_{K_1} \\&\qquad + \underbrace{\sum _{u\in V_1, v_1v_2\in E_2}^{} [\sigma (u,v_1)d(u,v_1)]^2 + [\sigma (u,v_2)d(u,v_2)]^2}_{K_2}.\\&\text { Now, } K_1\\&\quad = \sum _{u_1u_2\in E_1, v_1v_2\in E_2}^{} [\sigma (u_1,v_1)d(u_1,v_1)]^2 \\&\qquad + [\sigma (u_2,v_2)d(u_2,v_2)]^2\\&\quad \le \sum _{u_1u_2\in E_1, v_1v_2\in E_2}^{} [\sigma _1(u_1)d_1(u_1) + 2 \sigma _2(v_1)d_2(v_1)]^2 \\&\qquad + [\sigma _1(u_2)d_1(u_2) + 2 \sigma _2(v_2)d_2(v_2)]^2\\&\quad = \sum _{u_1u_2\in E_1, v_1v_2\in E_2}^{} [\{\sigma _1^2(u_1)d_1^2(u_1) + \sigma _1^2(u_2)d_1^2(u_2)\} \\&\qquad + 4 \{\sigma _2^2(v_1)d_2^2(v_1) + \sigma _2^2(v_2)d_2^2(v_2)\}\\&\qquad + 4 \{\sigma _1(u_1)d_1(u_1) + \sigma _1(u_2)d_1(u_2)\}\{\sigma _2(v_1)d_2(v_1) \\&\qquad + \sigma _2(v_2)d_2(v_2)\}]\\&\quad \le m_2EFI(G_1) + 4m_1EFI(G_2) + 4ZF_1^*(G_1)ZF_1^*(G_2). \\&\text { Again, } K_2\\&\quad = \sum _{u\in V_1, v_1v_2\in E_2}^{} [\sigma (u,v_1)d(u,v_1)]^2 \\&\qquad + [\sigma (u,v_2)d(u,v_2)]^2\\&\quad \le \sum _{u\in V_1, v_1v_2\in E_2}^{} [\sigma _1(u)d_1(u) + 2 \sigma _2(v_1)d_2(v_1)]^2 \\&\qquad + [\sigma _1(u)d_1(u) + 2 \sigma _2(v_2)d_2(v_2)]^2\\&\le \sum _{u\in V_1, v_1v_2\in E_2}^{} [2\sigma _1^2(u)d_1^2(u) + 4 \{\sigma _2^2(v_1)d_2^2(v_1)\\&\qquad + \sigma _2^2(v_2)d_2^2(v_2)\}\\&\qquad + 4 \sigma _1(u)d_1(u) \{\sigma _2(v_1)d_2(v_1) + \sigma _2(v_2)d_2(v_2)\}]\\&\quad \le 2m_2ZF_1(G_1) + 4n_1EFI(G_2) \\&\qquad + 4n_1\varDelta _1 FZ_1^*(G_2). \end{aligned}$$

Hence, The result follows.

Application of edge F-index in mathematical chemistry

To study the fruitfulness of a TI, we have to correlate this index with at least one physico-chemical characteristic of a chemical compound. Due to the International Academy of Mathematical Chemistry instruction, we studied regression analysis of that index with physico-chemical properties of chemical compounds. As octane isomers and alkanes are a large, diverse group for the preliminary testing of indices, they are helpful for such a type of investigation. We construct a fuzzy graph for these chemical compounds, where each carbon atom represents a vertex and the bond between two carbon atoms represents an edge. The formula defines the MV of vertices and edges are defined as

$$\begin{aligned} \sigma (v)&= \frac{\hbox { Atomic energy of}\ v}{\text {Maximum atomic energy among all vertices}}\\ \mu (uv)&= \frac{\hbox { Bond energy of the edge}\ uv}{\vee \{\sigma (u), \sigma (v)\}}. \end{aligned}$$

Note that the atomic energy of Carbon is 1086.5 kj/mol and the bond energy of C-C is 345 kj/mol. Hence, \( \sigma (C) = 1.00\) and \(\mu (C-C) = 0.32\). This index is studied to model some physico-chemical properties of octane isomers and alkanes.

QSPR analysis of edge F-index for octane isomers

Here 18 octane isomers are considered. For octane isomers, the below six properties are considered:

  1. (i)

    Heat capacity (HC) This is a physical property of a substance, which is the quantity of heat supplied to a body for a single change of its temperature. Here we have considered the heat capacity of octane isomers at a fixed pressure.

  2. (ii)

    Enthalpy of vaporization (EV) This is the amount of enthalpy that must be added to a liquid to convert an amount of matter into a gas. The enthalpy of evaporation is a function of pressure where that transformation takes place.

  3. (iii)

    Standard enthalpy of vaporization (SEV) This is the quantity of heat required to evaporate one unit of a liquid at a constant temperature.

  4. (iv)

    Entropy (E) Entropy is a measure of how much energy is spread between atoms and molecules in a process and can be defined in terms of the statistical potential of a system or in terms of other thermodynamic quantities.

  5. (v)

    Acentric factor (AF) This is a conceptual number introduced by Pitzer Kenneth in 1955, which proved to be very effective in describing matter. It has become a standard for the phase properties of single and pure components.

  6. (vi)

    Heat of vaporization (HV) Here, we have considered the heat of vaporization of octane isomers at a fixed temperature (25 \(^{\circ }\)C).

Table 1 The value of some properties and edge F-index for octane isomers
Full size table

The values of these properties for octane isomers are taken from http://www.moleculardescriptors.eu and listed in Table 1. Also, the value of the edge F-index for octane isomers is listed in Table 1. Now the regression (linear, quadratic, cubic, exponential, logarithmic, power) analysis is studied below:

  1. (i)

    Heat capacity (HC)

    $$\begin{aligned}&\text {Linear: } HC = 0.4933(EFI) + 22.485, |R| = 0.6247,\\&\text {Quadratic: } HC = -0.1144(EFI)^2 + 2.3471(EFI) \\&\quad + 15.272, |R|= 0.6630,\\&\text {Cubic: } HC = -0.0439(EFI)^3 + 0.9325(EFI)^2\\&\quad - 5.7446(EFI) + 35.542, |R| = 0.6780,\\&\text {Exponential: } HC = 22.763 e^{0.0185(EFI)}, |R| = 0.6214,\\&\text {Logarithmic: } HC = 3.9782ln(EFI) + 18.242,\\&|R| = 0.6420,\\&\text {Power: } HC = 19.398(EFI)^{0.1497}, |R| = 0.6399. \end{aligned}$$
Fig. 2
figure 2

The correlation of edge F-index with heat capacity for octane isomers

Full size image

Remark 1

As \(0.62 \le |R| \le 0.68\), this index may be applied to determine the heat capacity of octane isomers. As \(|R| \le 0.70\), we do not prefer to use this index to determine the heat capacity of octane isomers.

Fig. 3
figure 3

The correlation of edge F-index with enthalpy of vaporization for octane isomers

Full size image
  1. (ii)

    Enthalpy of vaporization (EV)

    $$\begin{aligned}&\text {Linear: } EV = -0.8458(EFI) + 76.231, |R| = 0.8768,\\&\text {Quadratic: } EV = 0.0874(EFI)^2 - 2.4428 (EFI) \\&\quad + 83.112, |R|= 0.9119,\\&\text {Cubic: } EV = -0.0001(EFI)^3 + 0.0905(EFI)^2\\&\quad - 2.4707 (EFI) + 83.192, |R| = 0.9119,\\&\text {Exponential: } EV = 76.594 e^{-0.012(EFI)}, |R| = 0.8810,\\&\text {Logarithmic: } EV = -7.502ln(EFI) + 84.876,\\&|R| = 0.9044,\\&\text {Power: } EV = 86.777(EFI)^{0.109}, |R| = 0.9061. \end{aligned}$$

Remark 2

As \(0.87 \le |R| \le 0.92,\) this index is applicable to determine the enthalpy of vaporization of octane isomers. For quadratic, cubic, logarithmic and power regressions, \(|R| \ge 0.90,\) implies these models are highly preferable to determine the enthalpy of vaporization of octane isomers.

Fig. 4
figure 4

The correlation of edge F-index with standard enthalpy of vaporization for octane isomers

Full size image
  1. (iii)

    Standard enthalpy of vaporization (SEV)

    $$\begin{aligned}&\text {Linear: } SEV = -0.1695(EFI) + 10.543,\\&|R| = 0.9288,\\&\text {Quadratic: } SEV = 0.0124(EFI)^2 - 0.3969 (EFI) \\&\quad + 11.522, |R|= 0.9477,\\&\text {Cubic: } SEV = -0.0019(EFI)^3 + 0.0659(EFI)^2 \\&\quad - 0.8793 (EFI) + 12.905, |R| = 0.9499,\\&\text {Exponential: } SEV = 10.66 e^{-0.019(EFI)}, |R| = 0.9333,\\&\text {Logarithmic: } SEV = -1.488ln(EFI) + 12.243,\\&|R| = 0.9483,\\&\text {Power: } SEV = 12.843(EFI)^{0.163}, |R| = 0.9496. \end{aligned}$$

Remark 3

As \(0.92 \le |R| \le 0.95,\) this index is applicable to determine the SEV of octane isomers. As \(|R| \ge 0.90,\) implies these models are highly preferable to determine the SEV of octane isomers.

Fig. 5
figure 5

The correlation of edge F-index with entropy for octane isomers

Full size image
  1. (iv)

    Entropy (E)

    $$\begin{aligned}&\text {Linear: } E = -2.054(EFI) + 122.58, |R| = 0.9551,\\&\text {Quadratic: } E = -0.0287(EFI)^2 - 1.5287 (EFI) \\&\quad + 120.32, |R|= 0.9558,\\&\text {Cubic: } E = -0.034(EFI)^3 + 0.9383(EFI)^2 \\&\quad - 10.261 (EFI) + 145.34, |R| = 0.9610,\\&\text {Exponential: } E = 124.37 e^{-0.02(EFI)}, |R| = 0.9538,\\&\text {Logarithmic: } E = -17.4ln(EFI) + 141.86,\\&|R| = 0.9410,\\&\text {Power: } E = 194.65(EFI)^{0.168}, |R| = 0.9367. \end{aligned}$$

Remark 4

As \(0.93 \le |R| \le 0.97,\) this index is applicable to determine the entropy of octane isomers. As \(|R| \ge 0.90,\) implies these models are highly preferable to determine the entropy of octane isomers.

Fig. 6
figure 6

The correlation of edge F-index with acentric factor for octane isomers

Full size image
  1. (v)

    Acentric factor (AF)

    $$\begin{aligned}&\text {Linear: } AF = -0.0162(EFI) + 0.4709, |R| = 0.9584,\\&\text {Quadratic: } AF = 0.0006(EFI)^2 - 0.0266 (EFI) \\&\quad + 0.5159, |R|= 0.9630,\\&\text {Cubic: } AF = -0.0002(EFI)^3 + 0.0051(EFI)^2 \\ {}&\quad - 0.0674 (EFI) + 0.6329, |R| = 0.9648,\\&\text {Exponential: } AF = 0.507 e^{-0.05(EFI)}, |R| = 0.9627,\\&\text {Logarithmic: } AF = -0.14ln(EFI) + 0.6285,\\&|R| = 0.9630,\\&\text {Power: } AF = 0.8168(EFI)^{0.427}, |R| = 0.9586. \end{aligned}$$

Remark 5

As \(0.95 \le |R| \le 0.97,\) this index is applicable to determine the AF of octane isomers. As \(|R| \ge 0.90,\) implies these models are highly preferable to determine the acentric factor of octane isomers.

Fig. 7
figure 7

The correlation of edge F-index with heat of vaporization for octane isomers

Full size image
  1. (vi)

    Heat of vaporization (HV):

    $$\begin{aligned}&\text {Linear: } HV = -0.7947(EFI) + 44.749, |R| = 0.9156,\\&\text {Quadratic: } HV = 0.0583(EFI)^2 - 1.7393 (EFI) \\&\quad + 48.424, |R|= 0.9214,\\&\text {Cubic: } HV = -0.0694(EFI)^3 + 1.7111 (EFI)^2 \\&\quad - 14.514 (EFI) + 80.427, |R| = 0.9437,\\&\text {Exponential: } HV = 45.283 e^{-0.021(EFI)}, |R| = 0.9172,\\&\text {Logarithmic: } HV = -6.294 ln(EFI) + 51.346,\\&|R| = 0.9241,\\&\text {Power: } HV = 53.737(EFI)^{-0.164}, |R| = 0.9247. \end{aligned}$$

Remark 6

As \(0.91 \le |R| \le 0.95,\) this index is applicable to determine the heat of vaporization of octane isomers. As \(|R| \ge 0.90,\) implies these models are highly preferable to determine the HV of octane isomers.

The correlations of this index with different properties of octane isomers are depicted in Figs. 2, 3, 4, 5, 6 and 7. Edge F-index is highly correlated with EV, SEV, entropy, acentric factor and heat of vaporization and less correlated with heat capacity for octane isomers.

QSPR analysis of edge F-index for alkanes

Here 67 alkanes are considered. For alkanes, the below properties are considered:

  1. (i)

    Melting point (MP) This is the temperature at which a substance changes from solid to liquid.

  2. (ii)

    Surface tension (ST) This is the tendency of the resting liquid surfaces to shrink to the lowest possible surface area. Here we have considered the surface tensions of alkanes at a fixed temperature (20 \(^{\circ }\)C).

  3. (iii)

    Critical pressure (CP) Critical pressure of a substance is the stress associated with the CP of a substance. The CP of a substance can be defined as the point on the scale of temperature and pressure where a liquid can coexist with its vapour.

  4. (iv)

    Critical temperature (CT) The critical temperature is the maximum temperature where a substance can exist as a liquid.

  5. (v)

    Heat of vaporization (HV) Here, we have considered the heat of vaporization of alkanes at a fixed temperature (25 \(^{\circ }\)C).

  6. (vi)

    Molar refraction (MR) It is a measure of the total polarization of a mole of a substance and depends on its temperature, refractive index and pressure. Here we have considered the molar refraction of alkanes at a fixed temperature (20 \(^{\circ }\)C).

  7. (vii)

    Molar volume (MV) This is equal to the molar mass per unit mass density. Here we have considered the molar volume of alkanes at a fixed temperature (20 \(^{\circ }\)C).

  8. (viii)

    Boiling point (BP) It is the temperature where the vapor pressure of a liquid is equal to the pressure surrounding the liquid and the liquid is converted to vapor.

The values of these properties for alkanes are taken from http://www.moleculardescriptors.eu and https://pubchem.ncbi.nlm.nih.gov and listed in Table 2. Also, the value of edge F-index for alkanes are listed in Table 2. Now the regression (linear, quadratic, cubic, exponential, logarithmic, power) analysis is studied below:

  1. (i)

    Melting point (MP)

    $$\begin{aligned}&\text {Linear: } MP = 2.3613(EFI) -127.86, |R| = 0.2518,\\&\text {Quadratic: } MP = -0.2081(EFI)^2 + 5.7461(EFI) \\&\quad - 139.92, |R| = 0.2651,\\&\text {Cubic: } MP = 0.1493(EFI)^3 - 3.938(EFI)^2 \\&\quad + 33.484(EFI) - 198.87, |R| = 0.3438,\\&\text {Logarithmic: } MP = 17.892ln(EFI) - 144.8,\\&|R| = 0.2814. \end{aligned}$$

Remark 7

As \(|R| \le 0.30,\) this index is inadequate to determine the melting point of alkanes.

  1. (ii)

    Surface tension (ST)

    $$\begin{aligned}&\text {Linear: } ST = 0.4069(EFI) + 17.579, |R| = 0.5152,\\&\text {Quadratic: } ST = - 0.0647 (EFI)^2 + 1.5064 (EFI) \\&\quad - 13.297, |R| = 0.5896,\\&\text {Cubic: } ST = 0.0081(EFI)^3 - 0.2772 (EFI)^2 \\&\quad + 3.2164 (EFI) + 9.1642, |R| = 0.6024,\\&\text {Exponential: } ST = 17.575e^{0.0206(EFI)}, |R| = 0.5046,\\&\text {Logarithmic: } ST = 3.3939ln(EFI) + 13.938,\\&\quad |R| = 0.5731,\\&\text {Power: } ST = 14.574(EFI)^{0.1734}, |R| = 0.5647. \end{aligned}$$

Remark 8

As \(0.50 \le |R| \le 0.61,\) this index may be applied to determine the surface tension of alkanes. As \(|R| \le 0.70\), we do not prefer to use this index to determine the surface tension of alkanes.

  1. (iii)

    Critical pressure (CP)

    $$\begin{aligned}&\text {Linear: } CP = -0.6071(EFI) + 31.551 , |R| = 0.5114 ,\\&\text {Quadratic: } CP = 0.1439 (EFI)^2 -2.9439 (EFI) \\&\quad + 40.052 , |R| = 0.6852 ,\\&\text {Cubic: } CP = -0.0187 (EFI)^3 + 0.611 (EFI)^2 \\&\quad -6.4379 (EFI) + 47.568 , |R| = 0.7192 ,\\&\text {Exponential: } CP = 31.301e^{-0.021(EFI)}, |R| = 0.5289,\\&\text {Logarithmic: } CP = -5.092 ln(EFI) +36.961 ,\\&|R| = 0.6294 ,\\&\text {Power: } CP = 37.576(EFI)^{ -0.173}, |R| = 0.6462 . \end{aligned}$$

Remark 9

As \(0.51 \le |R| \le 0.72,\) this index may be applied to determine the ST of alkanes. For cubic regression, \(|R| \ge 0.70,\) implies this model is preferable to determine the ST of alkanes.

  1. (iv)

    Critical temperature (CT)

    $$\begin{aligned}&\text {Linear: } CT = 11.816(EFI) + 190.78 , |R| = 0.6812 ,\\&\text {Quadratic: } CT = -1.5366 (EFI)^2 +36.763 (EFI) \\&\quad + 100.03 , |R| = 0.7583,\\&\text {Cubic: } CT = 0.1245(EFI)^3 -4.646 (EFI)^2\\&\quad + 60.025(EFI) + 49.989 , |R| = 0.7648,\\&\text {Exponential: } CT = 191.68e^{0.0477(EFI)}, |R| = 0.6488,\\&\text {Logarithmic: } CT = 88.647ln(EFI) +106.69,\\&\quad |R| = 0.7499 ,\\&{\textbf {Power: }} CT = 134.18(EFI)^{ 0.3662}, |R| = 0.7304. \end{aligned}$$

Remark 10

As \(0.64 \le |R| \le 0.77,\) this index may be applied to determine the critical temperature of alkanes. For quadratic, cubic and logarithmic regressions, \(|R| \ge 0.70,\) implies these models are preferable to determine the critical temperature of alkanes.

  1. (v)

    Heat of vaporization (HV)

    $$\begin{aligned}&\text {Linear: } HV = 0.979(EFI) + 30.66, |R| = 0.4458,\\&\text {Quadratic: } HV = -0.2051(EFI)^2 + 4.4695(EFI) \\&\quad + 17.061, |R| = 0.5525,\\&\text {Cubic: } HV = 0.0152(EFI)^3 -0.6044(EFI)^2\\&\quad +7.6803(EFI) +9.295, |R| = 0.5589,\\&\text {Exponential: } HV = 30.225e^{ 0.0287(EFI)}, |R| = 0.4287,\\&\text {Logarithmic: } HV = 8.4234ln(EFI) + 21.376,\\&|R| = 0.5109,\\&\text {Power: } HV = 23.058(EFI)^{0.2463}, |R| = 0.4969. \end{aligned}$$

Remark 11

As \( |R| \le 0.5\) for linear, exponential and power regressions, these models are inadequate to determine the heat of vaporization of alkanes. As \(0.50 \le |R| \le 0.56\) for quadratic, cubic and logarithm regressions, these models may use to determine the heat of vaporization of alkanes. But, as \( |R| \le 0.7\), we do not prefer to use this index to determine the surface tension of alkanes.

  1. (vi)

    Molar refraction (MR)

    $$\begin{aligned}&\text {Linear: } MR = 1.3995(EFI)+ 27.816, |R| =0.6528 ,\\&\text {Quadratic: } MR = -0.1931(EFI)^2 + 4.6847(EFI) \\&\quad +15.016, |R| = 0.7248,\\&\text {Cubic: } MR = 0.0154(EFI)^3 -0.5974(EFI)^2\\&\quad +7.9358(EFI) +7.1531, |R| = 0.7300,\\&\text {Exponential: } MR = 28.234e^{ 0.039(EFI)}, |R| = 0.6298,\\&\text {Logarithmic: } MR = 11.418ln(EFI) +15.829,\\&\quad |R| =0.7096,\\&{\textbf {Power: }} MR = 20.067(EFI)^{0.3219}, |R| = 0.6950. \end{aligned}$$

Remark 12

As \(0.62 \le |R| \le 0.73,\) this index may applicable to determine the molar refraction of alkanes. For quadratic, cubic and logarithmic regressions, \(|R| \ge 0.70,\) implies these models are preferable to determine the molar refraction of alkanes.

Table 2 The value of some properties and edge F-index for alkanes
Full size table
  1. (vii)

    Molar volume (MV)

    $$\begin{aligned}&\text {Linear: } MV = 4.4036(EFI) +125.77, |R| = 0.6113,\\&\text {Quadratic: } MV = -0.6906(EFI)^2 + 16.155 (EFI) \\&\quad + 79.989 , |R| = 0.6973,\\&\text {Cubic: } MV = 0.0543(EFI)^3 -2.1156(EFI)^2\\&\quad +27.613(EFI) +52.275, |R| = 0.7033,\\&\text {Exponential: } MV = 126.39e^{ 0.0294(EFI)}, |R| = 0.5934,\\&\text {Logarithmic: } MV = 36.431ln(EFI) +87.019,\\&\quad |R| = 0.6739,\\&\text {Power: } MV = 97.299(EFI)^{0.2445}, |R| = 0.6615. \end{aligned}$$

Remark 13

As \(0.59 \le |R| \le 0.71,\) this index may be applied to determine the molar volume of alkanes. For cubic regression, \(|R| \ge 0.70,\) implies this model is preferable to determine the molar volume of alkanes.

  1. (viii)

    Boiling point (BP)

    $$\begin{aligned}&\text {Linear: } BP = 8.9325(EFI) + 38.333, |R| =0.6290 ,\\&\text {Quadratic: } BP = -1.3837(EFI)^2 +31.397(EFI) \\&\quad -43.39, |R| = 0.7280,\\&\text {Cubic: } BP = 0.1307(EFI)^3 -4.6497(EFI)^2\\&\quad +55.83(EFI) -95.947, |R| = 0.7391 ,\\&\text {Logarithmic: } BP = 68.744ln(EFI)-28.744,\\&|R| = 0.7104. \end{aligned}$$

Remark 14

As \(0.62 \le |R| \le 0.74,\) this index may be applied to determine the boiling point of alkanes. For quadratic, cubic and logarithmic regressions, \(|R| \ge 0.70,\) implies these models are preferable to determine the boiling point of alkanes.

Hence edge F-index is correlated with critical pressure, critical temperature, molar refraction, molar volume and boiling point and is less correlated with surface tension and heat of vaporization for alkanes. Also, this index is inadequate to determine the melting point of alkanes.

Conclusion

The molecular descriptors are a very much useful tool in mathematical chemistry. The edge F-index is proposed for FGs here. Bounds of this index are calculated for fuzzy graphs. The maximal FG has been investigated concerning this index for a given set of vertices. Some relations of this index with the second Zagreb index and hyper-Zagreb index are established. For an isomorphic FGs, it is shown that the value of this index is the same. Bounds of this index for some FG operations are determined. Also, an application of the index in mathematical chemistry is studied. For this, 18 octane isomers and 67 alkanes are considered and analyzed the correlation between this index with some properties of the octane isomers and alkanes. From the correlation coefficient value, we obtained this index is highly correlated with EV, SEV, E, AF and heat of vaporization and less correlated with heat capacity for octane isomers. Also, this index is correlated with critical pressure, critical temperature, molar refraction, molar volume and boiling point and is less correlated with surface tension and heat of vaporization for alkanes. But, this index is inadequate to determine the melting point of alkanes. Also, there are many problems related to this index on FGs which are unsolved till now. Some of the problems are:

  1. (i)

    Find the minimal n-vertex FG concerning this index.

  2. (ii)

    Find the maximal n-vertex tree (fuzzy) concerning this index.

  3. (iii)

    Find the minimal n-vertex tree (fuzzy) concerning this index.