1. Introduction and Preliminary Results
The representation of a graph is expressed by numbers, polynomials, and matrices. Graphs have their own characteristics that may be calculated by topological indices, and under graph automorphism, the topology of graphs remains unchanged. Degree-based topological indices are exceptionally important in different classes of indices and take on a vital role in graphic theory and in particular in science.
Silicate is a chemical compound and has many commercial uses. It is used for the manufacture of different glass and ceramics organic compounds in large scale due to its cheapness and availability everywhere in the world. Silicates can be obtained from the Earth’s crust. In general, solid silicates are well-characterized and stable. Silicates like sodium orthosilicate and metasilicate, which have alkali cations and tiny or chain-like anions, are water soluble. When crystallised from a solution, they generate multiple solid hydrates. Water glass, which is made up of soluble sodium silicates and combinations, is a significant industrial and home chemical. For the construction of networks rhombus oxide and silicate, we refer the readers to 10. Rhombus silicate network and rhombus oxide network are shown in Figures 1 and 2, respectively.
In this article, is considered a network with a vertex set and an edge set of and is the degree of vertex . Let denote the sum of the degrees of all vertices adjacent to a vertex . Graovac et al. defined fifth M-Zagreb indices as polynomials for a molecular graph [1], and these are characterized as follows.
Let be a graph. Then,
V. R. Kulli [2], motivated by the above indices, described some new topological indices, and he named them as the fifth M-Zagreb indices of first and second type and fifth hyper-M-Zagreb indices of first and second type of a graph . They are defined as
They also define a new version of Zagreb index which is called as the third Zagreb index or fifth -Zagreb [3].
Corresponding to the above indices, he defined the general fifth -Zagreb polynomial and the general fifth -Zagreb polynomial of a molecular graph as
The fifth - and - Zagreb polynomials of a graph are defined as
The fifth and Zagreb polynomials of the graph are defined as
2. Main Results
We have studied the topological indices introduced by Kulli [2, 4] named as fifth M-Zagreb indices, fifth M-Zagreb polynomials, and index and computed exact formulae of these indices for rhombus-type silicate and oxide networks. Ali et al. studied degree-based topological indices for various networks [5–8]. For the basic notations and definitions, see [9–11].
2.1. Results for the Rhombus Type of Silicate Networks
In this section, we calculate degree-based topological indices of the dimension for rhombus-type silicate networks. In the following theorems, we compute M-Zagreb indices and polynomials.
Theorem 2.1.1. Let be the rhombus-type silicate network; then, the first and second fifth M-Zagreb indices are equal to
Proof. The outcome can be obtained by using the edge partition in Table 1.
By using equation [5],By doing some calculations, we obtainThus, from [6],By doing some calculations, we obtain
Theorem 2.1.2. Consider the rhombus-type silicate network for . Then, the first and second general fifth M-Zagreb indices are equal to
Proof. Let be the rhombus-type silicate network. Table 1 shows such an edge partition of . Thus, from [9], it follows thatBy using edge partitions in Table 1, we obtainBy doing some calculations, we haveFrom [12], we haveBy using edge partitions in Table 1, we obtainBy doing some calculations, we have
Theorem 2.1.3. Consider the rhombus-type silicate network for . Then, the first and second hyper-fifth M-Zagreb indices are equal to
Proof. Let be the rhombus type of silicate network. Table 1 shows such an edge partition of . Thus, from [13], it follows thatBy using edge partitions in Table 1, we obtainBy doing some calculations, we haveFrom [14], we haveBy using edge partitions in Table 1, we obtainBy doing some calculations, we have
Theorem 2.1.4. Consider the rhombus-type silicate network for . Then, the third M-Zagreb index is equal to
Proof. Let be the rhombus silicate network. Table 1 shows such an edge partition of . Thus, from [15], it follows thatBy using edge partitions in Table 1, we obtainBy doing some calculations, we have
Corresponding to the above indices, we are going to compute general fifth M-Zagreb polynomials for rhombus-type silicate network .
Theorem 2.1.5. Let be the first type of rhombus-type silicate network; then, general fifth M-Zagreb polynomials of first and second type are equal to
Proof. We obtain the outcome with the edge partition in Table 1. It follows from [1] thatBy doing some calculations, we obtainAlso, from [3],By making some calculations, we obtain
Corresponding to the above indices, we are going to compute fifth M-Zagreb polynomials for rhombus-type silicate network .
Theorem 2.1.6. Let be the rhombus type of silicate network; then, fifth M-Zagreb polynomials of first and second type are equal to
Proof. We obtain the outcome with the edge partition in Table 1. It follows from [16] thatBy doing some calculations, we obtainAlso, from [4],By making some calculations, we obtain
Theorem 2.1.7. Let be the rhombus-type silicate network; then, hyper-fifth M-Zagreb polynomials of first and second type are equal to
Proof. We obtain the outcome with the edge partition in Table 1. It follows from [2] thatBy doing some calculations, we obtainAlso, from [10],By making some calculations, we obtain
2.2. Results for the Rhombus Type of Oxide Networks
Now, we are calculating fifth M-Zagreb topological indices of the rhombus-type oxide network , where .
Theorem 2.2.1. Let be the rhombus-type silicate network; then, the first and second fifth M-Zagreb indices are equal to
Proof. The outcome can be obtained by using the edge partition in Table 2.
By using equation [5],By doing some calculations, we obtainThus, from [6],By doing some calculations, we obtain
Theorem 2.2.2. Consider the rhombus-type oxide network for . Then, the first and second general fifth M-Zagreb indices are equal to
Proof. Let be the rhombus-type oxide network. Table 2 shows such an edge partition of . Thus, from [9], it follows thatBy using edge partitions in Table 2, we obtainBy doing some calculations, we haveFrom [12], we haveBy using edge partitions in Table 2, we obtainBy doing some calculations, we have
Theorem 2.2.3. Consider the rhombus-type oxide network for . Then, the first and second hyper fifth M-Zagreb indices are equal to
Proof. Let be the rhombus-type oxide network. Table 2 shows such an edge partition of . Thus, from [13], it follows thatBy using edge partitions in Table 2, we obtainBy doing some calculations, we haveFrom [14], we haveBy using edge partitions in Table 2, we obtainBy doing some calculations, we have
Theorem 2.2.4. Consider the rhombus-type silicate network for . Then, the third M-Zagreb index is equal to
Proof. Let be the rhombus-type oxide network. Table 2 shows such an edge partition of . Thus, from [15], it follows thatBy using edge partitions in Table 2, we obtainBy doing some calculations, we have
Corresponding to the above indices, we are going to compute general fifth M-Zagreb polynomials for rhombus type of oxide network .
Theorem 2.2.5. Let be the rhombus-type oxide network; then, general fifth M-Zagreb polynomials of first and second type are equal to
Proof. We obtain the outcome with the edge partition in Table 1. It follows from [1] thatBy doing some calculations, we obtainAlso, from [3],By making some calculations, we obtain
Corresponding to the above indices, we are going to compute fifth M-Zagreb polynomials for rhombus-type oxide network .
Theorem 2.2.6. Let be the rhombus-type oxide network; then, fifth M-Zagreb polynomials of first and second type are equal to
Proof. We obtain the outcome with the edge partition in Table 2. It follows from [16] thatBy doing some calculations, we obtainAlso, from [4],By making some calculations, we obtain
Theorem 2.2.7. Let be the rhombus-type oxide network; then, hyper-fifth M-Zagreb polynomials of first and second type are equal to
Proof. We obtain the outcome with the edge partition in Table 2. It follows from [2] thatBy doing some calculations, we obtainAlso, from [10],By making some calculations, we obtain
3. Conclusion
In this study, we computed sum of degree-based indices for and graphs of rhombus oxide and silicate structures. We also computed certain sum of degree-based polynomials such as fifth M-Zagreb, fifth hyper M-Zagreb, and generalized fifth M-Zagreb indices for and graphs of rhombus oxide and silicate structures. These facts may be useful for people working in computer science and chemistry fields who encounter chemical networks. These results can also play a vital role in the determination of the significance of silicate and oxide networks. Like certain other topological indices, determining the representations of derived graphs like these is an open question.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This work was supported in part by the Natural Science Fund of Education Department of Anhui Province under Grant KJ2020A0478.