An investigation of edge F-index on fuzzy graphs and application in molecular chemistry

The molecular descriptors are a useful tool in the spectral graph, molecular chemistry and several fields of chemistry and mathematics. The edge F-index is proposed for fuzzy graphs (FGs) 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. 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 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.

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 π -electron. F-index has introduced by Fortula and Gutman [3] in 2015.
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 , μ) is called a fuzzy set where μ is a called membership function of S whose domain is X and co-domain is [0,1].
where V is called vertex set of the fuzzy graph with vertex membership function σ : V → [0, 1] and edge membership function Note that, the edge (x, y) and (y, x) are considered as same and sometimes it called the edge x y 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 ∈ V is defined as: d(v) = x∈V μ(xv). Let Δ and δ 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 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.
The definition of ZI of crisp graph are Second ZI: In [16], ZI for a FG is defined as Islam and Pal [7] modified the definition of first Zagreb index in FG as Another topological indices for FGs are

Edge F-index for a fuzzy graph
In 2015, Fortula and Gutman introduced the F-index for crisp graph as: For an edge uv ∈ E, suppose, 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 For an edge uv ∈ E, we define 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.
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. Then,

Example 1 Suppose G be a FG depicted in
Therefore, for a FG G, F I (G) = E F I (G), in general. Some bounds of this index are provided here.

Theorem 1 Suppose, G be a fuzzy graph. Then, E F I
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, E F I (H ) ≤ E F I (G).
Proof As H be a partial-FSG of G, As a FSG is also a partial-FSG, the above result is also true for a FSG. The edge F-index for isomorphic FGs are studied below.
Theorem 3 Suppose G 1 and G 2 be two isomorphic FGs.
Proof As G 1 and G 2 be two isomorphic FGs, there exist a bijective map ψ :

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. Proof For any two real numbers x and y, (1) The next theorem provides the relation this index and hyper-Zagreb index for FGs.

Theorem 5 Suppose G be a FG. Then, E F I (G) ≥ 1 2 H Z I (G).
Proof For any two real numbers x and y, from Eq. (1), we have In Eq.
The relation among edge F-index, hyper-ZI and second ZI for fuzzy graphs is determined below.

Edge F-index for fuzzy graph operations
In this section, edge F-index is studied for some FG operations.
Hence, the result follows.
Proof By Theorem 7, we have Also, the following are holds: From those results and Theorem 7, the result follows.
Then the edge index of Hence, the result follows.
Proof As G 1 + G 2 is join graph of G 1 and G 2 , then for Then, Then the edge F-index of G 1 + G 2 is Hence, the result follows. Then, Then the edge F-index of G 1 + G 2 is: Hence, the result follows.
Proof As G 1 ∪ G 2 is union of G 1 and G 2 , then for any Hence, the edge F-index for G 1 ∪ G 2 is: Proof As G 1 ⊗ G 2 is strong product graph of G 1 and G 2 , Hence, Therefore, the edge F-index of G 1 ⊗ G 2 is: Again, Hence, the result follows.
Therefore, the edge F-index for G 1 G 2 is: Hence, Therefore, the edge F-index for G 1 • G 2 is: 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 Note that the atomic energy of Carbon is 1086.5 kj/mol and the bond energy of C-C is 345 kj/mol. Hence, σ (C) = 1.00 and μ(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: (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. (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. (iii) Standard enthalpy of vaporization (SEV) This is the quantity of heat required to evaporate one unit of a liquid at a constant temperature. (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. (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. (vi) Heat of vaporization (HV) Here, we have considered the heat of vaporization of octane isomers at a fixed temperature (25 • C).
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:   Remark 5 As 0.95 ≤ |R| ≤ 0.97, this index is applicable to determine the AF of octane isomers. As |R| ≥ 0.90, implies these models are highly preferable to determine the acentric factor of octane isomers.

Remark 6
As 0.91 ≤ |R| ≤ 0.95, this index is applicable to determine the heat of vaporization of octane isomers. As |R| ≥ 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:  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 Findex for alkanes are listed in Table 2. Now the regression (linear, quadratic, cubic, exponential, logarithmic, power) analysis is studied below:

Remark 11
As |R| ≤ 0.5 for linear, exponential and power regressions, these models are inadequate to determine the heat of vaporization of alkanes. As 0.50 ≤ |R| ≤ 0.56 for quadratic, cubic and logarithm regressions, these models may use to determine the heat of vaporization of alkanes. But, as |R| ≤ 0.7, we do not prefer to use this index to determine the surface tension 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: (i) Find the minimal n-vertex FG concerning this index. (ii) Find the maximal n-vertex tree (fuzzy) concerning this index. (iii) Find the minimal n-vertex tree (fuzzy) concerning this index.