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 [58]. For the basic notations and definitions, see [911].

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