On ev-degree and ve-degree topological indices

Recently two new degree concepts have been defined in graph theory: ev-degree and ve-degree. Also the evdegree and ve-degree Zagreb and Randić indices have been defined very recently as parallel of the classical definitions of Zagreb and Randić indices. It was shown that ev-degree and ve-degree topological indices can be used as possible tools in QSPR researches . In this paper we define the ve-degree and ev-degree Narumi–Katayama indices, investigate the predicting power of these novel indices and extremal graphs with respect to these novel topological indices. Also we give some basic mathematical properties of ev-degree and ve-degree NarumiKatayama and Zagreb indices.

predicting power of these novel indices and extremal graphs with respect to these topological indices. Also we give some basic mathematical properties of ev-degree and ve-degree Zagreb indices. The authors investigated the relationship between the total π-electron energy on molecules and Zagreb indices [3].
For the details see the references [4][5][6]. Randić  We refer the interested reader to [8][9][10] and the references therein for the up to date arguments about the Randić index.
The forgotten topological index for a connected graph G defined as; It was showed in [11] 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 [11][12][13] and references therein.
In the 1980s, Narumi and Katayama considered the production of the degrees of vertices and named it the "simple topological index'' [14]. Later for this graph invariant, the name ''Narumi-Katayama index'' was used in [15][16][17]. The extremal graphs with respect to was studied by Gutman and Ghorbani [15] , Zolfi and Ashrafi [20]. Some relations between the Narumi-Katayama index and the first Zagreb index were introduced in the more recent paper [21].
Multiplicative versions of first Zagreb index of a connected graph was defined by Eliasi et. al. in [22] as; For detailed discussions of the multiplicative version of Zagreb indices, we refer the interested reader to [23] and the references cited therein.
In the following section, we will give basic definitions of ev-degree and ve-degree concepts, ve-degree and evdegree Zagreb indices and as well as the basic mathematical properties of these novel topological indices. And also we give the definitions of ev-degree and ve-degree Narumi-Katayama indices.

ve-degree and ev-degree concepts and corresponding topological indices
In this section we give the definitions of ev-degree and ve-degree concepts which were given by Chellali et al. in [1] and the definitions and properties of ev-degree and ve-degree topological indices.
The following theorem states the relationship between the first Zagreb index and the total ve-degree of a connected graph .
Theorem 2.7 [1] For any connected graph , where ( ) denotes the total number of triangles in .
In [1], the authors suggested the idea that to carry these novel degree concepts to mathematical chemistry. One of the present author following this suggestion defined ev-degree and ve-degree Zagreb indices and showed that these novel group Zagreb and Randić indices give better correlation than well-known topological indices such as; Wiener, Zagreb and Randić indices to modelling some physicochemical properties of octane isomers [2]. And now, we give the definitions and some basic mathematical properties of ev-degree and ve-degree Zagreb indices which were given in [2].

Definition 2.8 [2]
Let be a connected graph and ∈ ( ). The ev-degree Zagreb index of the graph is defined as;

Definition 2.9 [2]
Let be a connected graph and ∈ ( ). The first ve-degree Zagreb alpha index of the graph is defined as;

Definition 2.10 [2]
Let be a connected graph and ∈ ( ). The first ve-degree Zagreb beta index of the graph is defined as; . Definition 2.11 [2] Let be a connected graph and ∈ ( ). The second ve-degree Zagreb index of the graph is defined as;

Definition 2.12 [2] Let be a connected graph and
∈ ( ). The ve-degree Randić index of the graph is defined as; And now we restate the some basic properties of ev-degree and ve-degree Zagreb indices which were given in [2].
And now we give the definitions of ev-degree and ve-degree Narumi-Katayama indices for a graph G.

Definition 2.17
The -Narumi-Katayama index of a graph G is defined with the following equation If a graph has an isolated vertex, its = 0 which is the minimal value of . We take the graphs without isolated vertices in the following results which will be introduced in the section four.

Definition 2.18 The -Narumi-Katayama index of a graph G is defined with the following equation
In the next section we investigate the predicting power of these novel topological indices and after that we investigate some mathematical properties of these novel indices.
In this section we compare the Narumi-Katayama index and its corresponding versions of the ev-Narumi-Katayama and ve-Narumi-Katayama 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 0.8959 (see Table 2). Also we find the Narumi-Katayama index of octane isomers values at the www.moleculardescriptors.eu (see Table 3). We also calculate and show the ev-Narumi-Katayama and the ve-Narumi-Katayama indices of octane isomers values in Table 3.    Table 4).
The cross-correlation matrix of the indices are given in Table 5.

Main results
In this section, we firstly give some basic mathematical properties of ve-degree, ev-Narumi-Katayama and ve-Narumi-Katayama indices. And secondly we investigate certain mathematical properties of ev-degree and vedegree Zagreb indices. Proof. The second part of this equality were given in [1]. The first part comes from that since every triangle consists of three vertices and edges , we count every triangle exactly three times for each vertex. Since the total number of triangles in the graph G will not be changed, the desired result acquired easily. □   as desired. We assume that the claim is true for = and we will show that it is true = + 1.
So the proof ends. □ Theorem 4.12.
The -vertex unicyclic graph with the maximal is the + (depicted in Fig.1) such that (c) The -vertex bicyclic graph with the maximal is (depicted in Fig.1) such that ( ) =        ( ) = ( ).

5.Conclusion
In this study we defined ev-degree and ve-degree Narumi-Katayama indices and showed that these novel degree based topological indices can be used possible tools for QSPR researches. Also we investigated some basic mathematical properties of ev-degree and ve-degree Narumi-Katayama and Zagreb indices. It can be interesting to compute the exact value of ev-degree and ve-degree topological indices for some graph operations. It can also be interesting to investigate the ev-degree and ve-degree concepts for the other topological indices for further studies.