On Some Banhatti Indices of Triangular Silicate, Triangular Oxide, Rhombus Silicate and Rhombus Oxide Networks

Silicates are the largest, the most complicated and the most interesting class of minerals by far. Kulli introduced the Banhatti indices of a graph. In this paper the general K-Banhatti indices, first and second K-Banhatti indices, K hyper Banhatti indices and modified K Banhatti indices for triangular silicate network, triangular oxide network, rhombus oxide network and rhombus silicate networks are computed.


Introduction
Kulli introduced the first and second K-Banhatti indices of a graph in [1]. These are defined as, ∑ and ∑ where indicate that the vertex and edge are incident in . Kulli also defined some properties of these newly defined indices. The coindices of K-Banhatti indices were also defined in his work. Later kulli defined K hyper-Banhatti indices of V-Phenylenic nanotubes and nanotorus [2]. Gutman et al developed relations between Banhatti and Zagreb indices and discussed the lower and upper bounds for Banhatti indices of a connected graph in terms of Zagreb indices [3]. Kulli et al also computed Banhatti indices for certain families of benzenoid systems [4]. Moreover, Kulli introduced multiplicative hyper-Banhatti indices and coindices, Banhatti geometric-arithmetic index connectivity Banhatti indices for certain families of benzenoid systems [5][6][7].
Silicates are the largest, the most complicated and the most interesting class of minerals by far. SiO tetrahedron is the basic chemical unit of silicates. The silicates sheets are rings of tetrahedrons linked by shared oxygen nodes to other rings in two dimensional planes producing a sheet like structures. A silicate can be obtained by fusing a metal oxide or a metal carbonate with sand. Essentially every silicate contains SiO tetrahedron. The corner and the center vertices represent oxygen and silicon ions respectively. These vertices are called oxygen nodes and silicon node respectively. A Silicate network is obtained in different ways. Paul Manuel has constructed a silicate network from a honeycomb network. In Figure 1, SiO tetrahedra is shown where the corner and the center vertices represent oxygen and silicon ions respectively.
Computing topological indices in mathematical chemistry is an important branch. Topological index has become a very useful tool in the prediction of physio chemical and pharmacological properties of a compound.
The number of vertices and edges are the topological index molecular structure matters. The main ingredients of the molecular topological models are topological indices which are the topological characterization of molecules by means of numerical invariants. These models are instrumental in the discovery of new applications of molecules with specific chemical, pharmacological and biological properties. The first use of a topological index was made by the chemist Harold Wiener in 1947 [8]. These indices are used by various Rhombus Silicate and Rhombus Oxide Networks researchers in their studies. In recent years many researchers have worked on computing topological indices [9][10][11][12][13][14][15][16][17].
The general first and second K-Banhatti indices of a graph are defined as [7]: where ∈ .

Calculating Banhatti Indices for Silicate Network
A silicate network ! of dimension 4 is given in In a triangular oxide network, from level 4 there are 6 types of edges based on the degree of the vertices of each edge. In table 1 the edge degree partition of triangular silicate network ! of order 4 is shown.

Calculating Banhatti Indices for Triangular Oxide Network
A triangular oxide network of dimension ! denoted by HI ! is given in figure 2. The number of vertices and the number of edges in a triangular oxide network are given by In a triangular oxide network, from level 4 there are four types of edges based on the degree of the vertices of each edge given in table 2. In Table 2 the edge degree partition of triangular oxide network HI ! of order 4 is shown. In Figure 3, the rhombus silicate network of dimension 3 is presented. If we delete all the silicon ions from the rhombus silicate network, then we obtain rhombus oxide network as shown in Figure 4 of dimension 3.  There are three types of edges based on degree of end vertices. The edge degree partition of the rhombus oxide network, >?HI ! is given in Table 4.

Conclusion
In this paper we have computed the exact values of the general K-Banhatti indices, first and second K-Banhatti indices, K hyper Banhatti indices and modified K Banhatti indices for triangular silicate network, triangular oxide network, rhombus oxide network and rhombus silicate networks that will help to understand the physical features, chemical reactivities and biological activities of the triangular silicate network, triangular oxide network, rhombus oxide network and rhombus silicate networks. These results can also provide a significant determination in the pharmaceutical industry.