Predicting Some Physicochemical Properties of Octane Isomers: A Topological Approach Using ev-Degree and ve-Degree Zagreb Indices

Topological indices have important role in theoretical chemistry for QSPR researches. Among the all topological indices the Randić and the Zagreb indices have been used more considerably than any other topological indices in chemical and mathematical literature. Most of the topological indices as in the Randić and the Zagreb indices are based on the degrees of the vertices of a connected graph. Recently novel two degree concepts have been defined in graph theory; ev-degrees and ve-degrees. In this study we define ev-degree Zagreb index, ve-degree Zagreb indices and ve-degree Randić index by using these new graph invariants as parallel to their corresponding classical degree versions. We compare these new group ev-degree and ve-degree indices with the other well-known and most used topological indices in literature such as; Wiener, Zagreb and Randić indices by modelling some physicochemical properties of octane isomers. We show that the ev-degree Zagreb index, the ve-degree Zagreb and the ve-degree Randić indices give better correlation than Wiener, Zagreb and Randić indices to predict the some specific physicochemical properties of octanes. We investigate the relations between the second Zagreb index and ev-degree and ve-degree Zagreb indices and some mathematical properties of ev-degree and ve-degree Zagreb indices.

properties of molecules in QSPR/QSAR researches. Quantitative structure-property/activity relationships (QSPR/QSAR) studies have very important role in theoretical chemistry. Octane isomers have been used widely in QSPR studies. The role of octane isomers in QSPR studies, we refer the interested reader [1][2][3][4] and references therein. Among the all topological indices, Wiener, Randić and Zagreb indices are the most used topological indices in the chemical and mathematical literature so far.
Very recently, Chellali, Haynes, Hedetniemi and Lewis have published a seminal study: On ve-degrees and evdegrees in graphs [5]. The authors defined two novel degree concepts in graph theory; ev-degrees and ve-degrees and investigate some basic mathematical properties of both novel graph invariants with regard to graph regularity and irregularity [5]. After given the equality of the total ev-degree and total ve-degree for any graph, also the total ev-degree and the total ve-degree were stated as in relation to the first Zagreb index. It was proposed in the article that the chemical applicability of the total ev-degree (and the total ve-degree) could be an interesting problem in view of chemistry and chemical graph theory.
In this study we define ev-degree Zagreb index, ve-degree Zagreb indices and ve-degree Randić index by using these new graph invariants. We define these novel topological indices invariants as parallel to corresponding original definitions of based on classical degree concept. We compare these new group ev-degree, ve-degree Zagreb and ve-degree Randić indices with the other well-known and most used topological indices such as Wiener, Zagreb and Randić indices by modelling some physicochemical properties of octane isomers.

Preliminaries
In this section we give some basic and preliminary concepts which we shall use later. A graph = ( , ) consists of two nonempty sets and 2-element subsets of namely . The elements of are called vertices and the elements of are called edges. For a vertex , deg ( ) show the number of edges that incident to . The set of all vertices which adjacent to is called the open neighborhood of and denoted by ( ). If we add the vertex to ( ), then we get the closed neighborhood of , . For the vertices and , ( , ) denotes the distance between and which means that minimum number of edges between and . In [6], the Wiener index of a connected graph G, the first topological index, was defined as; In his study, Wiener used the total distance between all different atoms (vertices) of paraffin to predict boiling point. We refer the interested reader to [7][8][9] and the references therein for the detailed discussion of Wiener index.
The first and second Zagreb indices [10] defined as follows: The first Zagreb index of a connected graph , defined as; And the second Zagreb index of a connected graph , defined as; The authors investigated the relationship between the total π-electron energy on molecules and Zagreb indices [10]. For the details see the references [11][12][13]. Randić investigated the measuring the extent of branching of the carbon-atom skeleton of saturated hydrocarbons via Randić index [14]. The Randić index of a connected graph G defined as; We refer the interested reader to [15][16][17] and the references therein for the up to date arguments about the Randić index. And now we give the definitions of ev-degree and ve-degree concepts which were given by Chellali et al.
in [5]. We illustrate these new degree definitions for the vertices and edges of the graph which are shown in Figure 1. Definition 2.4 [5] Let be a connected graph and ∈ ( ) . The total ev-degree of the graph is defined as; And the total ve-degree of the graph is defined as; Observation 2.5 [5] For any connected graph G, The following theorem states the relationship between the first Zagreb index and the total ve-degree of a connected graph .
Theorem 2.6 [5] For any connected graph G, where ( ) denotes the total number of triangles in G.
We can restate the Theorem 2.1 for the trees which are acyclic and are not contain any triangles.

Corollary 2.7
For any tree , And from this last equality we naturally consider to apply these two novel degree concepts to chemical graph theory by introducing ev-degree and ve-degree Zagreb indices as well as ve-degree Randić index.

Definition 2.8
Let be a connected graph and ∈ ( ). The ev-degree Zagreb index of the graph is defined as;

Definition 2.9
Let be a connected graph and ∈ ( ). The first ve-degree Zagreb alpha index of the graph is defined as;

Definition 2.10
Let be a connected graph and ∈ ( ). The first ve-degree Zagreb beta index of the graph is defined as;

Definition 2.11
Let be a connected graph and ∈ ( ). The second ve-degree Zagreb index of the graph is defined as;

Definition 2.12
Let be a connected graph and ∈ ( ). The ve-degree Randić index of the graph is defined as; . Example 2. 13 We compute these novel topological indices for the graph G in the Example 2.3 (see Figure

Results and Discussions
In this section we compare all above mentioned old and new topological indices with each other by using strong correlation coefficients acquired from the chemical graphs of octane isomers. We get the experimental results at the www.moleculardescriptors.eu (see Table 1). The following physicochemical features have been modeled: • Entropy, • Acentric factor (AcenFac), • Enthalpy of vaporization (HVAP), • Standard enthalpy of vaporization (DHVAP).
We select those physicochemical properties of octane isomers for which give reasonably good correlations, i.e. the absolute value of correlation coefficients are larger than 0.8 except from the property HVAP (see Table 2). Also we find the Wiener index, the first Zagreb index, the second Zagreb index and the Randić indices of octane isomers values at the www.moleculardescriptors.eu (see Table 3). We also calculate and show the ev-degree Zagreb index, the ve-degree Zagreb indices and the ve-degree Randić index of octane isomers values in Table 3.   It can be seen from the Table 2 Table 4).

Entropy:
We can see from the Table 4 that the ve-degree Zagreb index (S) gives the highest square of correlation coefficient for entropy. After that the first ve-degree Zagreb alpha index (S α ), the first Zagreb index (M 1 ), the vedegree Randić index (R α ) and the second ve-degree Zagreb index (S µ ) give the highest square of correlation coefficients, respectively.

Acentric factor (AcenFac):
We can see from the Table 4 that the first ve-degree Zagreb alpha index (S α ) gives the highest square of correlation coefficient for Acentric factor. After that the first ve-degree Zagreb beta index (S β ) and the second Zagreb index (M 2 ) give the same value. And the the second ve-degree Zagreb index (S µ ), the ev-degree Zagreb index (S) and ve-degree Randić index (R α ) give the highest square of correlation coefficients, respectively.

Enthalpy of vaporization (HVAP):
It can be seen from the Table 4 that the Randić index (R) gives the the highest square of correlation coefficient for Enthalpy of vaporization. After that the first Zagreb index (M 1 ), the ev-degree Zagreb index (S) , the ve-degree Randić index (R α ) and the the first ve-degree Zagreb alpha index (S α ) give the highest square of correlation coefficients, respectively.

Standard enthalpy of vaporization (DHVAP):
We can observe from the Table 4  give the highest square of correlation coefficients, respectively.
And now we investigate the relations between the old topological indices and the novel topological indices. The correlation coefficients between the Wiener, Zagreb, Randić indices and the ev-degree and ve-degree indices are shown in Table 5. It can be shown from the Table 5 that the first ve-degree Zagreb beta index (S β ) gives the highest absolute value of correlation coefficient with the Wiener index. The ev-degree Zagreb index (S) gives the highest correlation coefficient with the first Zagreb index (M 1 ). The first ve-degree Zagreb beta index (S β ) gives the highest absolute value of correlation coefficient with the Randić index. And it is very surprisingly see that the correlation coefficient between the second Zagreb index (M 2 ) and the first ve-degree Zagreb beta index (S β ) is one. We can see from the Table 3 that ( ) = 2 ( ) for the molecular graphs of octane isomers. But we know that ( ) ≠ 2 ( ) from the Example 2.3. The following section we investigate the relation between the second Zagreb index and the first ve-degree Zagreb beta index.

Table 5. The correlation coefficients between old and corresponding novel topological indices
The cross correlation matrix of ev-degree and ve-degree indices are given in Table 6. It can be shown from the Table 6 that the minimum correlation efficient among the all ve-degree and ev-degree indices is 0.9676 which is indicate strong correlation among all these novel indices. From the above arguments, we can say that the ve-degree and ev-degree indices are possible tools for QSPR researches.

Lower and upper bounds of ev-degree and ve-degree Zagreb indices for general graphs
In this section are given the relations between second Zagreb index and ve-degree and ev-degree Zagreb indices.
And also fundamental mathematical properties of ev-degree and ve-degree Zagreb indices are given. Hence, . □ Before we give the following interesting theorem, we mention the forgotten topological index [10]. The forgotten topological index for a connected graph G defined as; It was showed in [18] that the predictive power of the forgotten topological index is very close to the first Zagreb index for the entropy and acentric factor. For further studies about the forgotten topological index we refer to the interested reader [18][19][20] and references therein.
And now we give the maximum and minimum graph classes with respect to ev-degree and ve-degree Zagreb indices.

Lower bound is achieved if and only if G is a path and upper bound is achieved if and only if G is a complete
graph.
Proof We get that = = | ( ) ∪ ( )| from the definition of ev-degree of any edge of G. | ( ) ∪ ( )| reaches its maximum value for the complete graphs and its minimum value for the path for an edge of G. There are − 3 edges with their ev-degrees equals 4 and 2 edges with their ev-degrees equals 3 for the n-vertex path.

Lower bound is achieved if and only if G is a path and upper bound is achieved if and only if G is a complete graph.
Proof The proof is similar the proof of Theorem 4.7. □ Theorem 4.10 Let T be a tree of order ≥ 5 vertices then;

Lower bound is achieved if and only if T is a path and upper bound is achieved if and only if T is a star.
Proof The proof is similar the proof of Theorem 4.8. □

Lower bound is achieved if and only if G is a path and upper bound is achieved if and only if G is a complete
graph.
Proof The proof is similar the proof of Theorem 4.7. □

Lower bound is achieved if and only if T is a path and upper bound is achieved if and only if T is a star.
Proof The proof is similar the proof of Theorem 4.8. □

Conclusion
We proposed novel topological indices based on ev-degree and ve-degree concept which have been defined very recently in graph theory. It has been shown that these indices can be used as predictive means in QSAR researches.
Predictive power of these indices have been tested on by using some physicochemical properties of octanes.
Acquired results show that the new ev-degree and ve-degree indices give somewhat better results by analogy wellknown Wiener, Zagreb and Randić indices. In addition, we investigated basic mathematical properties of these novel topological indices. We have found a lower and upper bounds for the simple connected graphs. It can be interesting to find the exact value of the ev-degree and ve-degree Zagreb indices of some graph operations such as; direct, Cartesian, corona, tensor, hierarchical and generalized hierarchical product of graphs for further studies.
It can also be interesting to investigate the relations between the ev-degree and ve-degree Zagreb indices and the other well-known topological indices.