On ve-Degree and ev-Degree Zagreb Indices of Titania Nanotubes

Titania nanotubes are among the most investigated nanomaterials relating to their common applications in the manufacturing of corrosion-resistant, gas sensing and catalytic molecules. Topological indices which are graph invariants derived from molecular graphs of molecules are used in QSPR researches for modelling physicochemical properties of molecules. Topological indices are important tools for determining the underlying topology of a molecule in view of theoretical chemistry. Most of the topological indices are defined by using classical degree concept of graph theory. Recently two novel degree concepts have been defined in graph theory: ve-degrees and ev-degrees. By using both novel graph invariants, as parallel to their classical degree versions, the ev-degree Zagreb index, the ve-degree Zagreb indices and the vedegree Randić index have been defined very recently. In this study the ev-degree Zagreb index, the ve-degree Zagreb indices and the ve-degree Randić index of titania nanotubes were computed.


Introduction
Graph theory which is one of the most important branch of applied mathematics and chemistry has many applications from the basic sciences to the engineering sciences especially for solving and modelling of real-world problems. Chemical graph theory is the common place for graph theory and chemistry. Topological indices are indispensable tools for QSPR researches in view of theoretical chemistry and chemical graph theory. Topological indices have been used more than seventy years predicting and modelling physicochemical properties of chemical substances.
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 the vertex is added to , then the closed neighborhood of , is got. For the vertices and , , denotes the distance between and which means that minimum number of edges between and .
The first distance based topological index is the Wiener index which was defined by H. Wiener to modelling the boiling points of paraffin molecules [1]. Wiener, computed all distances between all atoms (vertices) in the molecular graph of paraffin molecules and named this graph invariant as "path number". The Wiener index of a simple connected graph G defined as follows; Many years later the path number renamed as "Wiener index" to honor Professor Harold Wiener for valuable contribution to mathematical chemistry. In the same year, the fKrst degree based topologKcal Kndex was proposed by Platt for modelKng physKcal propertKes of alcanes [2]. The Platt Kndex of a sKmple connected graph G defKned as follows; After these both studies, approximately twenty five years later the well-known degree based Zagreb indices were defined by Gutman and Trinijastić to modelling π-electron energy of alternant carbons [3]. The first Zagreb index of a simple connected graph defined as; And the second Zagreb index of a simple connected graph defined as; An alternative definition of the first Zagreb index of a simple connected graph is given in the following formula: In 1975, Randić defined the "Randić index" [4] to modelling molecular branching of carbon skeleton atoms as follows; Among the all topological indices, the above mentioned topological indices have been used for QSPR researches more considerably than any other topological indices in chemical and mathematical literature. The interested reader are referred to the following citations for up to date information about these well-known and the most used topological indices [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].
Recently two new degree definitions in graph theory have been given by Chellali et al [30]. The authors of [30] found the close relationship between the first Zagreb index and the total ev-degrees (ve-degrees). It was suggested in [30] that these novel degree concepts may be used to define novel degree based topological indices. Following this suggestion, ev-degree and ve-degree topological indices have been defined in [31], [32], [33], [34]. These novel ev-degree and ve-degree topological indices have been showed that give considerably good correlations to predict some physicochemical properties of octane molecules than wellknown and above mentioned topological indices; Wiener, Zagreb and Randić indices.
The aim of this paper is to compute the exact values of evdegree and ve-degree topological indices of titania nanotubes.

Results and Discussions
In this section, the definitions of ev-degree and ve-degree concepts which were given by Chellali et al. in [30] and the definitions and properties of ev-degree and ve-degree topological indices which were given in [31] are introduced. After that, ev-degree and ve-degree topological indices of titania nanotubes are computed.
Where . , denotes the number of triangles contain the edge e.
Where . denotes the number of triangles contain the vertex v.
Definition 5. [30] 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 6. [30] For any connected graph , 0 , ( ) = 0 ( ).
The following theorem states the relationship between the first Zagreb index and the total ve-degree of a connected graph . . The ve-degree Randić index of the graph is defined as; And now, to compute the ev-degree and ve-degree Zagreb indices of titania nanotubes are given. The following Figure 1, the molecular graph of titania nanotubes are given.

Degrees deg(v) Number of vertex ve-degrees cv
2mn-2m 13 From the FKgure 1, Kt Ks got the followKng Table 3 whKch gKves the classKfKcatKon of edges Kn relatKon to theKr ve-degrees of end vertKces. Table 3. The Class f cat on of Edges n Relat on to The r ve-Degrees of End Vert ces.

Proof. From Equation 13
and Table 1, it can be directly wrote that;

√195
Proof. From Equation 17 and Table 3 it can be directly wrote that;

Conclusions
In this study, the exact values of newly defined ev-degree and ve-degree topological indices of titania nanotubes were computed. This calculation will help to predict and model some physicochemical, optical and biological properties of titania nanotubes. It can be interesting to compute the evdegree and ve-degree topological indices of some other nanotubes and networks for further studies.