On multiplicative degree based topological indices for planar octahedron networks

: Chemical graph theory is a branch of graph theory in which a chemical compound is presented with a simple graph called a molecular graph. There are atomic bonds in the chemistry of the chemical atomic graph and edges. The graph is connected when there is at least one connection between its vertices. The number that describes the topology of the graph is called the topological index. Cheminformatics is a new subject which is a combination of chemistry, mathematics and information science. It studies quantitative structure-activity (QSAR) and structure-property (QSPR) relationships that are used to predict the biological activities and properties of chemical compounds. We evaluated the second multiplicative Zagreb index, first and second universal Zagreb indices, first and second hyper Zagreb indices, sum and product connectivity indices for the planar octahedron network, triangular prism network, hex planar octahedron network, and give these indices closed analytical formulas.


Introduction
Topological indices, provided by graph theory, are a valuable tool. Cheminformatics is a modern academic area that combines the subjects of chemistry, mathematics, and information science. It studies the relationships between quantitative structure-activity (QSAR) and structureproperty (QSPR) used to suggest biological activities and chemical compound properties. That is why scholars around the world are extremely interested in it.
The molecular structures are those in which atoms are connected by covalent bonds. In graph theory, atoms are considered as vertices and covalent bonds are as edges. Chem-informatics is a new area of research in which the subjects chemistry, mathematics, and information science are combined.
At present, in the field computational chemistry topological indices have a rising interest which is actually associated to their use in nonempirical quantitative structure-property relationship and quantitative-structure activity relationship. Topological descriptor Top(G) may also be defined with the property of isomorphism i.e for every graph H isomorphic to G, Top(G) = Top(H). The idea of topological indices was first introduced by Weiner (1947) during the lab work on boiling point of paraffin and named this result as Path number which was later named as Weiner Index.
In this article, we consider the silicate structure (Manuel and Rajasingh, 2011) derived from the POH network, TP network, and hex POH network (Simon Raj and George, 2015). The method of drawing planar octahedron networks with dimension m is as follows: Step 1: Let take a silicate network with dimension m.
Step 2: At the middle of each triangular face, fix new vertices, and connect them to oxide vertices in the corresponding triangle face.   Step 3: Link all these new vertices of the centre that lie in the same silicate cell.
Step 4: The resulting graph is called the planar octahedron network for the m dimension as shown in Figure 1. Delete all silicon vertices. We can also create the triangular prism network as shown in Figure 2 and the hex POH network as shown in Figure 3.
In the graph theory, the number of edges which is occurrence to the vertex that is the degree of vertex of the graph. Let φ be a graph. Then second multiplicative Zagreb index (Gutman et al., 1975) can be defined as: First and second indices (Kulli, 2016) of a graph φ are defined as: The first and second universal Zagreb index (Kulli, 2016) defined as: where: α∈R.
The sum and product connectivity of Multiplicative indices (Kulli, 2016) defined as:

Results
We research the Zagreb indices with its types, such as the second Zagreb multiplicative index, the first Zagreb hyper index, the second Zagreb hyper index, the first and second Zagreb universal indices and the multiplicative indices sum and product connectivity for the planar octahedron network, triangular prism network, hex planar octahedron network.

Results for planar octahedron network POH(m)
In this section, we calculate edge partition of topological indices of the dimension m for the planar octahedron network. In the coming theorems, we compute some important indices for planar octahedron network.
Using Table 1, we have: Proof. Let φ 1 ≅ POH(m) network. We have to prove first universal Zagreb index and using Eq. 4: Using Table 1, we have:

(d a ,d b ) Number of edges
This value is what, we get after calculations: For second universal-Zegreb index and by using Eq. 5, we have: Using Table 1, we have: This value is what, we get after calculations: 11664  5184 ). Proof. Let φ 1 ≅ POH(m) network. We have to prove the sum and product connectivity of multiplicative indices and by using Eq. 6, we have: For product connectivity of multiplicative indices and by using Eq. 7, we have:  This value is what, we get after calculations: Proof. Let φ 2 ≅ TP network and by using Eq. 1, we have:  This value is what, we get after calculations:   Proof. Let φ 2 ≅ TP network. Using Eq. 4, we have: Using  For second universal Zegreb index and by using Eq. 5: Using Table 2, we have:  Proof. Let φ 2 ≅ TP network. We have to prove the sum and product connectivity of multiplicative indices. First prove sum connectivity of multiplicative indices and by using Eq. 6:  For product connectivity of multiplicative indices and by using Eq. 7, we have:

Results for hex POH network (m)
In this section, we calculate edge partition of topological indices of the dimension m for the hex POH network (m).
In the coming theorems, we compute some important indices for hex POH network (m). Proof. Let φ 3 ≅ hex POH(m) and by using Eq. 1, we have: Using This value is what, we get after calculations: Proof. Let φ 3 ≅ hex POH network and by using Eq. 2, we have: Using Proof. Let φ 3 ≅ hex POH network (m). We have to prove first universal Zagreb index and by using Eq. 4, we have: Using For second universal Zegreb index and by using Eq. 5, we have: Using  Proof. Let φ 3 ≅ hex POH(m). We have to prove the sum and product connectivity of multiplicative indices. First we prove sum connectivity of multiplicative indices and by using Eq. 6, we have:

Conclusions
In this paper, second multiplicative Zagreb index, first hyper Zagreb and second hyper Zagreb, first universal Zagreb and second hyper Zagreb, sum and product connectivity of multiplicative indices computed for the planar octahedron network, triangular prism network and hex planar octahedron network. Chemical point of view these results may be helpful for people working in computer science and chemistry who encounter hexderived networks. There exist many open problems for calculating the expressions of similar derived networks we are hoping to compute the other multiplicative degree based indices.