On Computation Degree-Based Topological Descriptors for Planar Octahedron Networks

School of Computer Engineering, Anhui Wonder University of Information Engineering, Hefei 231201, China Department of Mechanical Engineering, College of Engineering, Qassim University, Unaizah, Saudi Arabia Department of Mathematics, Riphah International University, Faisalabad, Pakistan Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Pakistan School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China


Introduction
Graph theory provides topological indices, which are a useful tool. Cheminformatics is a contemporary academic discipline that brings together chemistry, mathematics, and information science. It investigates the connections between quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR), which are used to predict biological activities and chemical compound characteristics. e silicate structures [1] formed from the POH network, TP network, and hex POH network [2] are discussed in this article. e following is the procedure for making POH networks .
Step 1: consider a m-dimensional silicate network.
Step 2: connect new vertices in the centre of each triangular face to existing vertices in the adjacent triangular face.
Step 3: all of the new centre vertices in the same silicate cell must be connected.
Step 4: for the m dimension, the resultant graph is known as the planar octahedron network as shown in Figure 1. Remove all silicon vertices from the graph. e triangle prism network as shown in Figure 2 and the hex POH network as shown in Figure 3 are also possible.
Let ψ represent a graph. en, modified first and second K-Banhatti indices [3] can be defined as Harmonic K-Banhatti index [4] of a graph ψ is defined as Symmetric division index of a graph [5] is defined as Augmented Zagreb index of a graph ψ [4] is defined as Inverse sum index of a graph ψ is defined as

Results for Planar Octahedron Network POH (m)
. e planar octahedron network is the resulting graph for the m dimension. All silicon vertices should be removed from the scene. ere are also the triangular prism network and the hex POH network. Now, we calculate several key indices for the POH network in the following theorems.
Proof. Let ψ 1 � POH(m). From equation (1), we have Using Table 1, we have We get the following value after calculations: Let ψ 1 � POH(m). From equation (2), we have Using Table 1, we have We get the following value after calculations: □ Theorem 2. e harmonic K-Banhatti and symmetric division indices are equal in the POH (m) network.
Proof. Let φ 1 � POH(m) network; from equation (3), Using Table 1, we have We get the following value after calculations:

Journal of Mathematics
For the symmetric division index of a graph using equation (3), Using Table 1, we have After calculations, □ Theorem 3. en, augmented Zagreb and inverse sum indices are equal to the POH network.
Proof. Let φ 1 � POH(m) network. For the augmented Zagreb index, using equation (5), (22) Using After some calculations, we get For the inverse sum index and by using equation (6), we have Using Table 1, we have After calculations, Proof. Let ψ 2 � TP(m). From equation (1), we have Using Table 2, we have After some calculations, we get Let ψ 2 � TP(m). From equation (2), we have Using Table 2, we have After calculations, Proof. Let φ 2 � TP(m) network, and by using equation (3), we have Using Table 2, we have After calculations, For the symmetric division index of a graph using equation (3), we have Using Table 2, we have After some calculations, we get Proof. Let φ 2 � POH(m) network. Using equation (5) for the augmented Zagreb index, (43) Using For the inverse sum index and by using equation (6), we have (46) Using Table 2, we have After calculations,

Results for Hexagonal Planar Octahedron (HPOH)
Network. In this part, we propose the theorem for the HPOH network.

Theorem 7. e first and second modified K-Banhatti indices are equal to the hex POH network:
Proof. Let ψ 3 � HPOH(m). From equation (1), we have (50) Using We get the following value after calculations: Let ψ 2 � HPOH(m). From equation (2), we have Using Table 3, we have Journal of Mathematics 7 We get the following value after calculations: □ Theorem 8. en, harmonic K-Banhatti and symmetric division indices are equal to the hex POH network: (56) Proof. Let φ 1 � HPOH(m) network, and from equation (3), Using Table 3, we have We get the following value after calculations: For the symmetric division index of a graph using equation (3), we have Using After calculations, □ Theorem 9. e augmented Zagreb and inverse sum indices are equal to the hex POH network: Proof. Let φ 1 � HPOH(m) network. Using equation (5) for the augmented Zagreb index, (64) Using We get the following value after calculations: For the inverse sum index and by using equation (6), we have

Conclusion
In this paper, first and the second K-Banhatti, harmonic K-Banhatti, symmetric division, augmented Zagreb, and inverse sum indices have been computed for the planar octahedron networks. From a chemical standpoint, these findings might be useful for computer scientists and chemists, who come across these networks. Additional multiplicative degree-based indices should be computed soon.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
Wang Zhen was responsible for software Parvez Ali contributed in collection of data. Haidar Ali contributed to original draft preparation. Ghulam Dustigeer was responsible for methodology. Jia-Bao Liu reviewed and edited the manuscript. Journal of Mathematics 11