Topological Indices of mth Chain Silicate Graphs

Abstract

A topological index is a numerical representation of a chemical structure, while a topological descriptor correlates certain physico-chemical characteristics of underlying chemical compounds besides its numerical representation. A large number of properties like physico-chemical properties, thermodynamic properties, chemical activity, and biological activity are determined by the chemical applications of graph theory. The biological activity of chemical compounds can be constructed by the help of topological indices such as atom-bond connectivity (ABC), Randić, and geometric arithmetic (GA). In this paper, Randić, atom bond connectivity (ABC), Zagreb, geometric arithmetic (GA), ABC₄, and GA₅ indices of the mth chain silicate $SL(m,n)$ network are determined.

1. Introduction and Preliminary Results

Graph theory is a branch of numerical science in which we apply the apparatuses of the diagram hypothesis to demonstrate the phenomena of compounds scientifically. In chemical graph theory, the edges represent the covalent bonding between atoms, and the vertices of a molecular graph represent atoms. The significance of the molecular graph is that the hydrogen atom is omitted from it.

“A topological index is a quantity that is somehow calculated from the molecular graph and for which we believe that it reflects relevant structural features of the underlying molecule” [1]. Until the late 1970s, topological indices only had information on atom connectivity, but in this age, they have many properties of saturated hydrocarbons [2]. The whole theory of topological indices was started by Wiener [3] when he was doing the experiment on the boiling point of paraffin.

As a consequence of various experiments, we can claim that topological indices are useful in QSPR/QSAR studies. The correlation between physico-chemical properties (QSPR) and the biological activity relationships (QSAR) of the molecules was tested with the help of topological indices. It is pertinent to note that the topological indices have some major classes such as counted related topological indices, degree-based topological indices, and distance-based topological indices of graphs. The degree-based topological indices are very important in chemical graph theory to test the attributes of compounds and drugs, which have been mostly used in chemical and pharmacy engineering [4,5].

In 1975, Milan Randić put forward the topological index, which is based on degree, in his seminar paper [6]. His index was denoted by $R_{-\frac{1}{2}}\left(G\right)$ and defined as:

$$R_{-\frac{1}{2}}\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}\frac{1}{\sqrt{deg\left(u\right)deg\left(v\right)}}$$

(1)

Randić named it the “branching index”, but then it was re-named [7,8] to the “connectivity index”. Currently, it is called the “Randić index” [9]. Let $R_{\alpha }\left(G\right)$ be the general Randić index defined as:

$$R_{\alpha }\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}{\left(deg\left(u\right)deg\left(v\right)\right)}^{\alpha }\mathrm{for}\alpha =1,\frac{1}{2},-1,-\frac{1}{2}$$

(2)

“Ruder Boskovic” is the group for theoretical chemistry at the Institute of Zagreb, and Balaban et al. was a member of this group. Therefore, he named the $M_1$ and $M_2$ topological indices as “Zagreb group indices”. Later on, the “Zagreb group index” was abbreviated to “Zagreb index”, and now, $M_1$ is renamed as the “first Zagreb index”; on the other hand, $M_2$ is called the “second Zagreb index”. These indices are defined as:

$$M_1\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}(deg\left(u\right)+deg\left(v\right))$$

(3)

$$M_2\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}\left(deg\left(u\right)deg\left(v\right)\right)$$

(4)

A new topological index that is obtained by the extension of Equation (1), introduced by Emesto Estrada, is called the “atom-bond connectivity index” and defined as:

$$\mathrm{ABC}\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}\sqrt{\frac{deg\left(u\right)+deg\left(v\right)-2}{deg\left(u\right)deg\left(v\right)}}$$

(5)

The ABC₄ index is the fourth version of the atom-bond connectivity index. The ABC₄ index was formulated by Ghorbani et al. [10] and is defined as:

$${\mathrm{ABC}}_4\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}\sqrt{\frac{S_u+S_v-2}{S_u{S}_v}}$$

(6)

Another topological index invented by Vukičević and Furtula [11] was named the “geometric-arithmetic index”. It is defined as:

$$\mathrm{GA}\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}\frac{2\sqrt{deg\left(u\right)deg\left(v\right)}}{\left(deg\right(u)+deg(v\left)\right)}$$

(7)

GA₅ is the fifth version of the geometric arithmetic index [12]. The GA₅ index was proposed by Graovac et al. [13] and is defined as:

$${\mathrm{GA}}_5\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}\frac{2\sqrt{S_u{S}_v}}{(S_u+S_v)}$$

(8)

In this article, graph G is to be taken as a graph with edge set $E\left(G\right)$ and vertex set $V\left(G\right)$, and the degree of vertex $u\in V\left(G\right)$ is denoted as $d_u$ and $S_u={\displaystyle \sum _{v\in N_G\left(u\right)}}{S}_v$ where $N_G\left(u\right)=\{v\in V\left(G\right)\mid uv\in E\left(G\right)\}$.

Hayat et al. worked on the degree-based topological index, such as for silicate, hexagonal, honeycomb, and oxide [14]. For more conclusive results related to topological indices of chemical graphs and their graph invariants, see [15,16,17,18,19,20,21,22,23].

Construction of the Silicate Chain Graph:

SiO₄ tetrahedra are found nearly in all the silicates. Silicates are immensely essential and complicated minerals. We get silicates from metal carbonates with sand or from fusing metal oxides. Silicates behave as the building blocks of the usual rock-forming minerals.

Consider a single tetrahedron (i.e., a pyramid having a triangular base). Place oxygen atoms at the four corners of a tetrahedron, and the silicon atom is bonded with equally-spaced atoms of oxygen. The resulting tetrahedron is a silicate tetrahedron, which is shown in Figure 1a, and when this tetrahedron joins with other tetrahedra linearly, then a single-row silicate chain is formed, as shown in Figure 1b. When two tetrahedra join together corner-to-corner, then each tetrahedron shares its oxygen atom with the other tetrahedron, as shown in Figure 1c. After this sharing, these two tetrahedra can be joined with two other tetrahedra, as in Figure 1d. Now, extend this structure in one direction, then double the silicate chain formed, as in Figure 1e, were m is the number of row lines and n is the number of edges in a row line.

2. Results for mth Chain Silicate

In this section, we calculate the closed results for topological indices, which are based on vertex degrees of the mth silicate chain. We compute the general Randić for $\alpha =1,-1,\frac{1}{2},-\frac{1}{2}$, ABC, GA, ABC₄, and GA₅ indices for the mth chain silicate in this section. The number of edges of the mth chain silicate are $6mn$. In the following theorem, the general Randić index for the mth chain silicate is computed.

Theorem 1.

The general Randić index for $\mathcal{G}\cong SL(m,n)$ is equal to,

when $m=n$:

$$R_{\alpha }\left(\mathcal{G}\right)=\left\{\begin{array}{cc}36(3mn-6n+2)+18(3mn+3n-4)+9(3m+2),\hfill & \alpha =1;\hfill \\ 6(3mn-6n+2)+3\sqrt{2}(3mn+3n-4)+3(3m+2),\hfill & \alpha =\frac{1}{2};\hfill \\ \frac{1}{36}(3mn-6n+2)+\frac{1}{18}(3mn+3n-4)+\frac{1}{9}(3m+2),\hfill & \alpha =-1;\hfill \\ \frac{1}{6}(3mn-6n+2)+\frac{3mn+3n-4}{3\sqrt{2}}+\frac{1}{3}(3m+2),\hfill & \alpha =-\frac{1}{2}.\hfill \end{array}\right.$$

when $m<n$:

Case 1: When m is odd and n is even and vice versa. For $m>1$,

(i) when m is odd and n is even:

$$R_{\alpha }\left(\mathcal{G}\right)=\left\{\begin{array}{cc}36(3mn-2n-4m+2)+18(3mn+2n+m-5)+9(3m+3),\hfill & \alpha =1;\hfill \\ 6(3mn-2n-4m+2)+3\sqrt{2}(3mn+2n+m-5)+3(3m+3),\hfill & \alpha =\frac{1}{2};\hfill \\ \frac{1}{36}(3mn-2n-4m+2)+\frac{1}{18}(3mn+2n+m-5)+\frac{1}{9}(3m+3),\hfill & \alpha =-1;\hfill \\ \frac{1}{6}(3mn-2n-4m+2)+\frac{3mn+2n+m-5}{3\sqrt{2}}+\frac{1}{3}(3m+3),\hfill & \alpha =-\frac{1}{2}.\hfill \end{array}\right.$$

(ii) when m is even and n is odd:

$$R_{\alpha }\left(\mathcal{G}\right)=\left\{\begin{array}{cc}36(3mn-2n-4m+2)+18(3mn+2n+m-8)+9(3m+6),\hfill & \alpha =1;\hfill \\ 6(3mn-2n-4m+2)+3\sqrt{2}(3mn+2n+m-8)+3(3m+6),\hfill & \alpha =\frac{1}{2};\hfill \\ \frac{1}{36}(3mn-2n-4m+2)+\frac{1}{18}(3mn+2n+m-8)+\frac{1}{9}(3m+6),\hfill & \alpha =-1;\hfill \\ \frac{1}{6}(3mn-2n-4m+2)+\frac{3mn+m+2n-8}{3\sqrt{2}}+\frac{1}{3}(3m+6),\hfill & \alpha =-\frac{1}{2}.\hfill \end{array}\right.$$

Case 2: When m and n both are even:

$$R_{\alpha }\left(\mathcal{G}\right)=\left\{\begin{array}{cc}36(3mn-2n-4m+2)+18(3mn+2n+m-6)+9(3m+4),\hfill & \alpha =1;\hfill \\ 6(3mn-2n-4m+2)+3\sqrt{2}(3mn+2n+m-6)+3(3m+4),\hfill & \alpha =\frac{1}{2};\hfill \\ \frac{1}{36}(3mn-2n-4m+2)+\frac{1}{18}(3mn+2n+m-6)+\frac{1}{9}(3m+4),\hfill & \alpha =-1;\hfill \\ \frac{1}{6}(3mn-2n-4m+2)+\frac{3mn+2n+m-6}{3\sqrt{2}}+\frac{1}{2}(3m+4),\hfill & \alpha =-\frac{1}{2}.\hfill \end{array}\right.$$

Case 3: When m and n both are odd:

$$R_{\alpha }\left(\mathcal{G}\right)=\left\{\begin{array}{cc}36(3mn-6m-2)+18(3mn+3m-2)+9(3m+4),\hfill & \alpha =1;\hfill \\ 6(3mn-6m-2)+3\sqrt{2}(3mn+3m-2)+3(3m+4),\hfill & \alpha =\frac{1}{2};\hfill \\ \frac{1}{36}(3mn-6m-2)+\frac{1}{18}(3mn+3m-2)+\frac{1}{9}(3m+4),\hfill & \alpha =-1;\hfill \\ \frac{1}{6}(3mn-6m-2)+\frac{3mn+3m-2}{3\sqrt{2}}+\frac{1}{3}(3m+4),\hfill & \alpha =-\frac{1}{2}.\hfill \end{array}\right.$$

when $m>n$:

Case 1: When m is odd and n is even and vice versa. For $m>2$,

(i) when m is odd and n is even:

$$R_{\alpha }\left(\mathcal{G}\right)=\left\{\begin{array}{cc}36(3mn-4n-2m)+18(3mn+4n-m-1)+9(3m+1),\hfill & \alpha =1;\hfill \\ 6(3mn-4n-2m)+3\sqrt{2}(3mn+4n-m-1)+3(3m+1),\hfill & \alpha =\frac{1}{2};\hfill \\ \frac{1}{36}(3mn-4n-2m)+\frac{1}{18}(3mn+4n-m-1)+\frac{1}{9}(3m+1),\hfill & \alpha =-1;\hfill \\ \frac{1}{6}(3mn-4n-2m)+\frac{(3mn+4n-m-1}{3\sqrt{2}}+\frac{1}{3}(3m+1),\hfill & \alpha =-\frac{1}{2}.\hfill \end{array}\right.$$

(ii) when m is even and n is odd:

$$R_{\alpha }\left(\mathcal{G}\right)=\left\{\begin{array}{cc}36(3mn-4n-2m)+18(3mn+4n-m-4)+9(3m+4),\hfill & \alpha =1;\hfill \\ 6(3mn-4n-2m)+3\sqrt{2}(3mn+4n-m-4)+3(3m+4),\hfill & \alpha =\frac{1}{2};\hfill \\ \frac{1}{36}(3mn-4n-2m)+\frac{1}{18}(3mn+4n-m-4)+\frac{1}{9}(3m+4),\hfill & \alpha =-1;\hfill \\ \frac{1}{6}(3mn-4n-2m)+\frac{3mn+4n-m-4}{3\sqrt{2}}+\frac{1}{3}(3m+4),\hfill & \alpha =-\frac{1}{2}.\hfill \end{array}\right.$$

Case 2: When m and n both are even:

$$R_{\alpha }\left(\mathcal{G}\right)=\left\{\begin{array}{cc}36(3mn-4n-2m-2)+18(3mn+4n-m+2)+27m,\hfill & \alpha =1;\hfill \\ 6(3mn-4n-2m-2)+3\sqrt{2}(3mn+4n-m+2)+9m,\hfill & \alpha =\frac{1}{2};\hfill \\ \frac{1}{36}(3mn-4n-2m-2)+\frac{1}{18}(3mn+4n-m+2)+\frac{m}{3},\hfill & \alpha =-1;\hfill \\ \frac{1}{6}(3mn-4n-2m-2)+\frac{3mn+4n-m+2}{3\sqrt{2}}+m,\hfill & \alpha =-\frac{1}{2}.\hfill \end{array}\right.$$

Case 3: When m and n both are odd. For $n>1$:

$$R_{\alpha }\left(\mathcal{G}\right)=\left\{\begin{array}{cc}36(3mn-4n-2m-2)+18(3mn+4n-m+2)+27m,\hfill & \alpha =1;\hfill \\ 6(3mn-4n-2m-2)+3\sqrt{2}(3mn+4n-m+2)+9m,\hfill & \alpha =\frac{1}{2};\hfill \\ \frac{1}{36}(3mn-4n-2m-2)+\frac{1}{18}(3mn+4n-m+2)+\frac{m}{3},\hfill & \alpha =-1;\hfill \\ \frac{1}{6}(3mn-4n-2m-2)+\frac{3mn+4n-m+2}{3\sqrt{2}}+m,\hfill & \alpha =-\frac{1}{2}.\hfill \end{array}\right.$$

Proof. 

When $m=n$, $m>1$:

Let $\mathcal{G}$ be the chain silicate Figure 2, based on degree n. The $E\left(\mathcal{G}\right)$ can be divided into three partitions. The $E_1\left(\mathcal{G}\right)$ have $3m+2$ edges $uv$; where $deg\left(u\right)=deg\left(v\right)=3$. The $E_2\left(\mathcal{G}\right)$ have $3mn+3n-4$ edges $uv$, where $deg\left(u\right)=3$ and $deg\left(v\right)=6$. The $E_3\left(\mathcal{G}\right)$ have $3mn-6n+2$ edges $uv$; where $deg\left(u\right)=deg\left(v\right)=6$. See

Table 1. Edge partition of silicate chain ($SL(m,n)$), when $m=n$, based on the degrees of the end vertices of each edge.

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(3,3)$	$3m+2$
$(3,6)$	$3mn+3n-4$
$(6,6)$	$3mn-6n+2$

for the edge partitions of $\mathcal{G}$. Thus, from Equation (2), it follows that:

$$R_{\alpha }\left(\mathcal{G}\right)={\displaystyle \sum _{uv\in E\left(\mathcal{G}\right)}}{\left(deg\left(u\right)deg\left(v\right)\right)}^{\alpha }$$

For $\alpha =1$:

$R_{\alpha }\left(\mathcal{G}\right)$ is the general Randić index, defined by:

$$R_1\left(\mathcal{G}\right)=\sum _{j=1}^3{\displaystyle \sum _{uv\in E_j\left(G\right)}}deg\left(u\right)·deg\left(v\right)$$

By using Table 1, we get the following:

$$R_1\left(\mathcal{G}\right)=9|E_1\left(\mathcal{G}\right)|+18|E_2\left(\mathcal{G}\right)|+36|E_3\left(\mathcal{G}\right)|$$

$$⟹R_1\left(\mathcal{G}\right)=36(3mn-6n+2)+18(3mn+3n-4)+9(3m+2).$$

For $\alpha =\frac{1}{2}$:

We apply the formula of $R_{\alpha }\left(\mathcal{G}\right)$.

$$R_{\frac{1}{2}}\left(\mathcal{G}\right)=\sum _{j=1}^3{\displaystyle \sum _{uv\in E_j\left(G\right)}}\sqrt{deg\left(u\right)·deg\left(v\right)}$$

By using Table 1, we get:

$$R_{\frac{1}{2}}\left(\mathcal{G}\right)=3|E_1\left(\mathcal{G}\right)|+2\sqrt{3}|E_2\left(\mathcal{G}\right)|+6|E_3\left(\mathcal{G}\right)|$$

$$⟹R_{\frac{1}{2}}\left(\mathcal{G}\right)=6(3mn-6n+2)+3\sqrt{2}(3mn+3n-4)+3(3m+2)$$

For $\alpha =-1$:

We apply the formula of $R_{\alpha }\left(\mathcal{G}\right)$.

$$R_{-1}\left(\mathcal{G}\right)=\sum _{j=1}^3{\displaystyle \sum _{uv\in E_j\left(\mathcal{G}\right)}}\frac{1}{deg\left(u\right)·deg\left(v\right)}$$

$$R_{-1}\left(\mathcal{G}\right)=\frac{1}{9}|E_1\left(\mathcal{G}\right)|+\frac{1}{18}|E_2\left(\mathcal{G}\right)|+\frac{1}{36}\left|E_3\left(\mathcal{G}\right)\right|$$

$$⟹R_{-1}\left(\mathcal{G}\right)=\frac{1}{36}(3mn-6n+2)+\frac{1}{18}(3mn+3n-4)+\frac{1}{9}(3m+2)$$

For $\alpha =-\frac{1}{2}$

We apply the formula of $R_{\alpha }\left(\mathcal{G}\right)$.

$$R_{-\frac{1}{2}}\left(\mathcal{G}\right)=\sum _{j=1}^3{\displaystyle \sum _{uv\in E_j\left(\mathcal{G}\right)}}\frac{1}{\sqrt{deg\left(u\right)·deg\left(v\right)}}$$

$$R_{-\frac{1}{2}}\left(\mathcal{G}\right)=\frac{1}{3}|E_1\left(\mathcal{G}\right)|+\frac{1}{3\sqrt{2}}|E_2\left(\mathcal{G}\right)|+\frac{1}{6}\left|E_3\left(\mathcal{G}\right)\right|$$

$$⟹R_{-\frac{1}{2}}\left(\mathcal{G}\right)=\frac{1}{6}(3mn-6n+2)+\frac{3mn+3n-4}{3\sqrt{2}}+\frac{1}{3}(3m+2)$$

When $m<n$:

The proof of the following cases, (i) when m is odd and n is even, (ii) when m is even and n is odd, (iii) when m and n both are even, and (iv) when m and n both are odd, is the same as $m=n$ by using

Table 2. Edge partition of silicate chain ($SL(m,n)$), when $m<n$ and m is odd, based on the degrees of the end vertices of each edge.

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(3,3)$	$3m+3$
$(3,6)$	$3mn+m+2n-5$
$(6,6)$	$3mn-2n-4m+2$

,

Table 3. Edge partition of silicate chain ($SL(m,n)$), when $m<n$ and m is even, based on the degrees of the end vertices of each edge.

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(3,3)$	$3m+6$
$(3,6)$	$3mn+m+2n-8$
$(6,6)$	$3mn-2n-4m+2$

,

Table 4. Edge partition of silicate chain ($SL(m,n)$), when $m<n$, m and n both are even, based on the degrees of the end vertices of each edge.

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(3,3)$	$3m+4$
$(3,6)$	$3mn+m+2n-6$
$(6,6)$	$3mn-4m-2n+2$

and

Table 5. Edge partition of silicate chain ($SL(m,n)$), when $m<n$, m and n both are odd, based on the degrees of the end vertices of each edge.

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(3,3)$	$3m+4$
$(3,6)$	$3mn+3m-2$
$(6,6)$	$3mn-6m-2$

, respectively.

When $m>n$:

The proof of the following cases, (i) when m is odd and n is even, (ii) when m is even and n is odd, (iii) when m and n both are even, and (iv) when m and n both are odd, is the same as $m=n$ by using

Table 6. Edge partition of silicate chain ($SL(m,n)$), when $m>n$ and m is odd, based on the degrees of end vertices of each edge.

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(3,3)$	$3m+1$
$(3,6)$	$3mn-m+4n-1$
$(6,6)$	$3mn-2m-4n$

,

Table 7. Edge partition of silicate chain ($SL(m,n)$), when $m>n$ and m is even, based on the degrees of the end vertices of each edge.

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(3,3)$	$3m+4$
$(3,6)$	$3mn-m+4n-4$
$(6,6)$	$3mn-2m-4n$

,

Table 8. Edge partition of silicate chain ($SL(m,n)$), when $m>n$, m and n both are even, based on the degrees of the end vertices of each edge.

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(3,3)$	$3m$
$(3,6)$	$3mn-m+4n+2$
$(6,6)$	$3mn-2m-4n-2$

and

Table 9. Edge partition of silicate chain ($SL(m,n)$), when $m>n$, m and n both are odd, based on the degrees of the end vertices of each edge.

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(3,3)$	$3m$
$(3,6)$	$3mn-m+4n+2$
$(6,6)$	$3mn-2m-4n-2$

, respectively.  □

In this theorem, we find the result of the first Zagreb index for chain silicate.

Theorem 2.

For the chain silicate $\mathcal{G}$, the first Zagreb index is equal to,

when $m=n$:

$$M_1\left(\mathcal{G}\right)=12(3mn-6n+2)+9(3mn+3n-4)+6(3m+2)$$

when $m<n$:

Case 1: When m is odd and n is even or vice versa. For $m>1$,

(i) when m is odd and n is even:

$$M_1\left(\mathcal{G}\right)=12(3mn-2n-4m+2)+9(3mn+2n+m-5)+6(3m+3)$$

(ii) when m is even and n is odd:

$$M_1\left(\mathcal{G}\right)=12(3mn-2n-4m+2)+9(3mn+2n+m-8)+6(3m+6)$$

Case 2: When m and n both are even.

$$M_1\left(\mathcal{G}\right)=12(3mn-2n-4m+2)+9(3mn+2n+m-6)+6(3m+4)$$

Case 3: When m and n both are odd.

$$M_1\left(\mathcal{G}\right)=12(3mn-6m-2)+9(3mn+3m-2)+6(3m+4)$$

when $m>n$:

Case 1: When m is odd and n is even or vice versa. For $m>2$,

(i) when m is odd and n is even:

$$M_1\left(\mathcal{G}\right)=12(3mn-4n-2m)+9(3mn+4n-m-1)+6(3m+1)$$

(ii) when m is even and n is odd:

$$M_1\left(\mathcal{G}\right)=12(3mn-4n-2m)+9(3mn+4n-m-4)+6(3m+4)$$

Case 2: When m and n both are even.

$$M_1\left(\mathcal{G}\right)=12(3mn-4n-2m-2)+9(3mn+4n-m+2)+18m$$

Case 3: When m and n both are odd.

$$M_1\left(\mathcal{G}\right)=12(3mn-4n-2m-2)+9(3mn+4n-m+2)+18m$$

Proof. 

When $m=n$:

Let $\mathcal{G}$ denotes the chain silicate. By using the edge partition from Table 1, the result follows. From Equation (3), we have:

$$M_1\left(\mathcal{G}\right)={\displaystyle \sum _{uv\in E\left(\mathcal{G}\right)}}(deg\left(u\right)+deg\left(v\right))=\sum _{j=1}^3{\displaystyle \sum _{uv\in E_j\left(\mathcal{G}\right)}}(deg\left(u\right)+deg\left(v\right))$$

$$M_1\left(\mathcal{G}\right))=6|E_1\left(\mathcal{G}\right)|+9|E_2\left(\mathcal{G}\right)|+12|E_3\left(\mathcal{G}\right)|$$

By doing some calculation, we get the following:

$$⟹M_1\left(\mathcal{G}\right)=12(3mn-6n+2)+9(3mn+3n-4)+6(3m+2)$$

When $m<n$:

The proof of the following cases, (i) when m is odd and n is even, (ii) when m is even and n is odd, (iii) when m and n both are even, and (iv) when m and n both are odd, is the same as $m=n$ by using Table 2, Table 3, Table 4 and Table 5, respectively.

When $m>n$:

The proof of the following cases, (i) when m is odd and n is even, (ii) when m is even and n is odd, (iii) when m and n both are even, and (iv) when m and n both are odd, is the same as $m=n$ by using Table 6, Table 7, Table 8 and Table 9, respectively.  □

Corollary 1.

$M_2\left(\mathcal{G}\right)$ is always equal to $R_1\left(\mathcal{G}\right)$.

Now, we calculate the results of the following topological indices of chain silicate.

Theorem 3.

Let $\mathcal{G}$ be the chain silicate, based on degree n, then we have,

when $m=n$:

• $ABC\left(\mathcal{G}\right)$ = $\frac{1}{3}\sqrt{\frac{5}{2}}(3mn-6n+2)+\frac{1}{3}\sqrt{\frac{7}{2}}(3mn+3n-4)+\frac{2}{3}(3m+2)$. For $m>1$;
• GA$\left(\mathcal{G}\right)$ = $6mn+3m-3n$. For $m>1$;
• ${ABC}_4\left(\mathcal{G}\right)$ = $\frac{1}{15}\sqrt{\frac{29}{2}}(3mn-16m+20)+\frac{2}{15}\sqrt{7}(3m-4)+\frac{1}{3}\sqrt{\frac{23}{30}}(3mn-9n-4m+14)+\frac{2}{27}\sqrt{13}(2m-8)+\frac{7m}{9\sqrt{2}}+\frac{1}{6}\sqrt{\frac{37}{10}}(4m+4)+\frac{1}{9}\sqrt{\frac{43}{6}}(2m+2n-10)+\frac{1}{9}\sqrt{\frac{11}{2}}(3m+3n-14)+\frac{2}{9}\sqrt{2}(4m+4n-20)+\frac{\sqrt{17}}{2}+\frac{\sqrt{\frac{23}{2}}}{6}+\frac{\sqrt{\frac{13}{5}}}{3}+\frac{\sqrt{\frac{10}{3}}}{3}$. For $m>3$;
• ${GA}_5\left(\mathcal{G}\right)$ = $3mn+\frac{1}{4}\sqrt{15}(3mn-9n-4m+14)+\frac{24\sqrt{2}m}{17}+\frac{4}{13}\sqrt{10}(4m+4)+\frac{2}{5}\sqrt{6}(2m+2n-10)+\frac{6}{19}\sqrt{10}(3m+3n-14)+\frac{3}{7}\sqrt{5}(4m+4n-20)-11m+4\sqrt{2}+\frac{8\sqrt{3}}{7}+\frac{8\sqrt{5}}{9}+16$. For $m>3$.

when $m<n$:

Case 1: When m is odd and n is even or vice versa.

(i) when m is odd and n is even:

• $ABC\left(\mathcal{G}\right)$ = $\frac{1}{3}\sqrt{\frac{5}{2}}(3mn-2n-4m+2)+\frac{1}{3}\sqrt{\frac{7}{2}}(3mn+2n+m-5)+\frac{2}{3}(3m+3)$. For $m>1$
• GA$\left(\mathcal{G}\right)$ = $6mn$. For $m>1$
• ${ABC}_4\left(\mathcal{G}\right)$ = $\frac{1}{15}\sqrt{\frac{29}{2}}(3mn-12n-4m+26)+\frac{2}{15}\sqrt{7}(3m-3)+\frac{1}{3}\sqrt{\frac{23}{30}}(3mn-7n-6m+16)+\frac{2}{27}\sqrt{13}(m+n-7)+\frac{7(m+n-1)}{18\sqrt{2}}+\frac{1}{6}\sqrt{\frac{37}{10}}(2m+2n+2)+\frac{1}{9}\sqrt{\frac{43}{6}}(m+3n-11)+\frac{1}{9}\sqrt{\frac{11}{2}}(m+5n-17)+\frac{2}{9}\sqrt{2}(m+7n-23)+\frac{\sqrt{17}}{2}+\sqrt{\frac{11}{2}}+\frac{\sqrt{\frac{23}{2}}}{6}+\frac{\sqrt{\frac{13}{5}}}{3}+\frac{\sqrt{\frac{10}{3}}}{3}$. For $m>2$
• ${GA}_5\left(\mathcal{G}\right)$ = $3mn+\frac{1}{4}\sqrt{15}(3mn-7n-6m+16)+\frac{12}{17}\sqrt{2}(m+n-1)+\frac{4}{13}\sqrt{10}(2m+2n+2)+\frac{2}{5}\sqrt{6}(m+3n-11)+\frac{6}{19}\sqrt{10}(m+5n-17)+\frac{3}{7}\sqrt{5}(m+7n-23)-11n+4\sqrt{2}+\frac{8\sqrt{3}}{7}+\frac{8\sqrt{5}}{9}+24$. For $m>2$

(ii) when m is even and n is odd:

• $ABC\left(\mathcal{G}\right)$ = $\frac{1}{3}\sqrt{\frac{5}{2}}(2-4m+3mn-2n)+\frac{1}{3}\sqrt{\frac{7}{2}}(3mn+2n+m-8)+\frac{2}{3}(3m+6)$. For $m>1$
• GA$\left(\mathcal{G}\right)$ = $6mn$. For $m>1$
• $AB{C}_4\left(G\right)$ = $\frac{1}{15}\sqrt{\frac{29}{2}}(3mn-12n-4m+26)+\frac{2}{15}\sqrt{7}(3m-6)+\frac{1}{3}\sqrt{\frac{23}{30}}(3mn-8n-5m+18)+\frac{2}{27}\sqrt{13}(m+n-7)+\frac{7(m+n-1)}{18\sqrt{2}}+\frac{1}{6}\sqrt{\frac{37}{10}}(m+3n-11)+\frac{1}{9}\sqrt{\frac{43}{6}}(m+3n-11)+\frac{1}{9}\sqrt{\frac{11}{2}}(m+5n-17)+\frac{2}{9}\sqrt{2}(m+7n-23)+\sqrt{17}+\sqrt{22}+\frac{2\sqrt{\frac{13}{5}}}{3}+\frac{2\sqrt{\frac{10}{3}}}{3}$. For $m>2$
• ${GA}_5\left(\mathcal{G}\right)$ = $3mn+\frac{1}{4}\sqrt{15}(3mn-8n-5m+18)+\frac{12}{17}\sqrt{2}(m+n-1)+\frac{4}{13}\sqrt{10}(m+3n-11)+\frac{2}{5}\sqrt{6}(m+3n-11)+\frac{6}{19}\sqrt{10}(m+5n-17)+\frac{3}{7}\sqrt{5}(m+7n-23)-11n+8\sqrt{2}+\frac{16\sqrt{3}}{7}+\frac{16\sqrt{5}}{9}+25$. For $m>2$

Case 2: When m and n both are even.

• $ABC\left(\mathcal{G}\right)$ = $\frac{1}{3}\sqrt{\frac{5}{2}}(3mn-2n-4m+2)+\frac{1}{3}\sqrt{\frac{7}{2}}(3mn+2n+m-6)+\frac{2}{3}(4+3m)$
• GA$\left(\mathcal{G}\right)$ = $6mn$
• ${ABC}_4\left(\mathcal{G}\right)$ = $\frac{1}{15}\sqrt{\frac{29}{2}}(3mn-16m+8)+\frac{2}{15}\sqrt{7}(3m-2)+\frac{1}{3}\sqrt{\frac{23}{30}}(3mn-16n+3m+36)+\frac{2}{27}\sqrt{13}(m+n-6)+\frac{7(m+n-2))}{18\sqrt{2}}+\frac{1}{6}\sqrt{\frac{37}{10}}(m+3n-2)+\frac{1}{9}\sqrt{\frac{43}{6}}(4m-6)+\frac{1}{9}\sqrt{\frac{11}{2}}(m+5n-20)+\frac{2}{9}\sqrt{2}(2m+6n-24)+\frac{\sqrt{17}}{2}+\sqrt{\frac{11}{2}}+\frac{\sqrt{\frac{23}{2}}}{6}+\frac{\sqrt{\frac{13}{5}}}{3}+\frac{\sqrt{\frac{10}{3}}}{3}$. For $m>2$
• ${GA}_5\left(\mathcal{G}\right)$ = $3mn+\frac{1}{4}\sqrt{15}(3mn-16n+3m+36)+\frac{12}{17}\sqrt{2}(m+n-2)+\frac{4}{13}\sqrt{10}(m+3n-2)+\frac{2}{5}\sqrt{6}(4m-6)+\frac{6}{19}\sqrt{10}(m+5n-20)+\frac{3}{7}\sqrt{5}(2m+6n-24)-12m+n+4\sqrt{2}+\frac{8\sqrt{3}}{7}+\frac{8\sqrt{5}}{9}+8$. For $m>2$

Case 3: When m and n both are odd.

• $ABC\left(\mathcal{G}\right)$ = $\frac{1}{3}\sqrt{\frac{5}{2}}(3mn-6m-2)+\frac{1}{3}\sqrt{\frac{7}{2}}(3mn+3m-2)+\frac{2}{3}(3m+4)$
• GA$\left(\mathcal{G}\right)$ = $6mn$
• ${ABC}_4\left(\mathcal{G}\right)$ = $\frac{1}{15}\sqrt{\frac{29}{2}}(3mn-20n+4m+48)+\frac{2}{15}\sqrt{7}(3m-2)+\frac{1}{3}\sqrt{\frac{23}{30}}(3mn-13m+4)+\frac{2}{27}\sqrt{13}(m+n-6)+\frac{7(m+n-2))}{18\sqrt{2}}+\frac{1}{6}\sqrt{\frac{37}{10}}(m+3n-2)+\frac{1}{9}\sqrt{\frac{43}{6}}(m+3n-12)+\frac{1}{9}\sqrt{\frac{11}{2}}(m+5n-20)+\frac{2}{9}\sqrt{2}(m+7n-26)+\frac{\sqrt{17}}{2}+\sqrt{\frac{11}{2}}+\frac{\sqrt{\frac{23}{2}}}{6}+\frac{\sqrt{\frac{13}{5}}}{3}+\frac{\sqrt{\frac{10}{3}}}{3}$. For $m>1$
• ${GA}_5\left(\mathcal{G}\right)$ = $3mn+\frac{1}{4}\sqrt{15}(3mn-13m+4)+\frac{12}{17}\sqrt{2}(m+n-2)+\frac{4}{13}\sqrt{10}(m+3n-2)+\frac{2}{5}\sqrt{6}(m+3n-12)+\frac{6}{19}\sqrt{10}(m+5n-20)+\frac{3}{7}\sqrt{5}(m+7n-26)+8m+4\sqrt{2}+\frac{8\sqrt{3}}{7}+\frac{8\sqrt{5}}{9}+48$. For $m>1$

when $m>n$:

Case 1: When m is odd and n is even or vice versa.

(i) when m is odd and n is even:

• $ABC\left(\mathcal{G}\right)$ = $\frac{1}{3}\sqrt{\frac{5}{2}}(3mn-4n-2m)+\frac{1}{3}\sqrt{\frac{7}{2}}(3mn+4n-m-1)+\frac{2}{3}(3m+1)$. For $m>2$
• GA$\left(\mathcal{G}\right)$ = $6mn$. For $m>2$
• ${ABC}_4\left(\mathcal{G}\right)$ = $\frac{1}{15}\sqrt{\frac{29}{2}}(3mn-14n-2m+12)+\frac{2}{15}\sqrt{7}(m+2n-3)+\frac{1}{3}\sqrt{\frac{23}{30}}(3mn-5m-8n+11)+\frac{2}{27}\sqrt{13}(m+n-9)+\frac{7(m+n+1)}{18\sqrt{2}}+\frac{1}{6}\sqrt{\frac{37}{10}}(m+3n+7)+\frac{1}{9}\sqrt{\frac{43}{6}}(m+3n-9)+\frac{1}{9}\sqrt{\frac{11}{2}}(m+5n-11)+\frac{2}{9}\sqrt{2}(m+7n-17)+\frac{\sqrt{17}}{2}+\sqrt{\frac{11}{2}}+\frac{\sqrt{\frac{23}{2}}}{6}+\frac{\sqrt{\frac{13}{5}}}{3}+\frac{\sqrt{\frac{10}{3}}}{3}$. For $m>4$
• ${GA}_5\left(\mathcal{G}\right)$ = $3mn+\frac{1}{4}\sqrt{15}(3mn-8n-5m+11)+\frac{12}{17}\sqrt{2}(m+n+1)+\frac{4}{13}\sqrt{10}(m+3n+7)+\frac{2}{5}\sqrt{6}(m+3n-9)+\frac{6}{19}\sqrt{10}(m+5n-11)+\frac{3}{7}\sqrt{5}(m+7n-17)-11n+4\sqrt{2}+\frac{8\sqrt{3}}{7}+\frac{8\sqrt{5}}{9}+8$. For $m>4$

(ii) when m is even and n is odd:

• $ABC\left(\mathcal{G}\right)$ = $\frac{1}{3}\sqrt{\frac{5}{2}}(3mn-4n-2m)+\frac{1}{3}\sqrt{\frac{7}{2}}(3mn+4n-m-4)+\frac{2}{3}(3m+4)$. For $m>2$
• GA$\left(\mathcal{G}\right)$ = $6mn$. For $m>2$
• ${ABC}_4\left(\mathcal{G}\right)$ = $\frac{1}{15}\sqrt{\frac{29}{2}}(3mn-14n-2m+12)+\frac{2}{15}\sqrt{7}(m+2n-6)+\frac{1}{3}\sqrt{\frac{23}{30}}(3mn-8n-5m+12)+\frac{2}{27}\sqrt{13}(m+n-9)+\frac{7(m+n+1)}{18\sqrt{2}}+\frac{1}{6}\sqrt{\frac{37}{10}}(m+3n-5)+\frac{1}{9}\sqrt{\frac{43}{6}}(m+3n-9)+\frac{1}{9}\sqrt{\frac{11}{2}}(m+5n-11)+\frac{2}{9}\sqrt{2}(m+7n-17)+\sqrt{17}+\sqrt{22}+\frac{2\sqrt{\frac{13}{5}}}{3}+\frac{2\sqrt{\frac{10}{3}}}{3}$. For $m>4$
• ${GA}_5\left(\mathcal{G}\right)$ = $3mn+\frac{1}{4}\sqrt{15}(3mn-8n-5m+12)+\frac{12}{17}\sqrt{2}(m+n+1)+\frac{4}{13}\sqrt{10}(m+3n-5)+\frac{2}{5}\sqrt{6}(m+3n-9)+\frac{6}{19}\sqrt{10}(m+5n-11)+\frac{3}{7}\sqrt{5}(m+7n-17)-11n+4\sqrt{2}+\frac{16\sqrt{3}}{7}+\frac{16\sqrt{5}}{9}+9$. For $m>4$

Case 2: When m and n both are even.

• $ABC\left(\mathcal{G}\right)$ = $\frac{1}{3}\sqrt{\frac{5}{2}}(3mn-4n-2m-2)+\frac{1}{3}\sqrt{\frac{7}{2}}(3mn+4n-m+2)+2m$. For $m>2$
• GA$\left(\mathcal{G}\right)$ = $6mn+4$. For $m>2$
• ${ABC}_4\left(\mathcal{G}\right)$ = $\frac{1}{15}\sqrt{\frac{29}{2}}(3mn-14n-2m+4)+\frac{2}{15}\sqrt{7}(3m-6)+\frac{1}{3}\sqrt{\frac{23}{30}}(3mn-6n-7m+12)+\frac{2}{27}\sqrt{13}(m+n-10)+\frac{7(m+n+2)}{18\sqrt{2}}+\frac{1}{6}\sqrt{\frac{37}{10}}(m+3n+10)+\frac{1}{9}\sqrt{\frac{43}{6}}(m+3n-8)+\frac{1}{9}\sqrt{\frac{11}{2}}(m+5n-8)+\frac{2}{9}\sqrt{2}(m+7n-14)+\frac{\sqrt{17}}{2}+\sqrt{\frac{11}{2}}+\frac{\sqrt{\frac{23}{2}}}{6}+\frac{\sqrt{\frac{13}{5}}}{3}+\frac{\sqrt{\frac{10}{3}}}{3}$. For $m>6$
• ${GA}_5\left(\mathcal{G}\right)$ = $3mn+\frac{1}{4}\sqrt{15}(3mn-6n-7m+12)+\frac{12}{17}\sqrt{2}(m+n+2)+\frac{4}{13}\sqrt{10}(m+3n+10)+\frac{2}{5}\sqrt{6}(m+3n-8)+\frac{6}{19}\sqrt{10}(m+5n-8)+\frac{3}{7}\sqrt{5}(m+7n-14)-13n+2m+4\sqrt{2}+\frac{8\sqrt{3}}{7}+\frac{8\sqrt{5}}{9}-4$. For $m>6$

Case 3: When m and n both are odd.

• $ABC\left(\mathcal{G}\right)$ = $\frac{1}{3}\sqrt{\frac{5}{2}}(3mn-4n-2m-2)+\frac{1}{3}\sqrt{\frac{7}{2}}(3mn+4n-m+2)+2m$. For $m>3$ and $n>1$
• GA$\left(\mathcal{G}\right)$ = $6mn+4$. For $m>3$ and $n>1$
• ${ABC}_4\left(\mathcal{G}\right)$ = $\frac{1}{15}\sqrt{\frac{29}{2}}(3mn-12n-4m+8)+\frac{2\sqrt{7}n}{5}+\frac{1}{3}\sqrt{\frac{23}{30}}(3mn-11n-2m+2)+\frac{2}{27}\sqrt{13}(m+n-10)+\frac{7(m+n+2)}{18\sqrt{2}}+\frac{1}{6}\sqrt{\frac{37}{10}}(m+3n+10)+\frac{1}{9}\sqrt{\frac{43}{6}}(m+3n-8)+\frac{1}{9}\sqrt{\frac{11}{2}}(m+5n-8)+\frac{2}{9}\sqrt{2}(m+7n-14)+\frac{\sqrt{17}}{2}+\sqrt{\frac{11}{2}}+\frac{\sqrt{\frac{23}{2}}}{6}+\frac{\sqrt{\frac{13}{5}}}{3}+\frac{\sqrt{\frac{10}{3}}}{3}$. For $m>5$
• ${GA}_5\left(\mathcal{G}\right)$ = $3mn+\frac{1}{4}\sqrt{15}(3mn-11n-2m+2)+\frac{12}{17}\sqrt{2}(m+n+2)+\frac{4}{13}\sqrt{10}(m+3n+10)+\frac{2}{5}\sqrt{6}(m+3n-8)+\frac{6}{19}\sqrt{10}(m+5n-8)+\frac{3}{7}\sqrt{5}(m+7n-14)-3m-8n+4\sqrt{2}+\frac{8\sqrt{3}}{7}+\frac{8\sqrt{5}}{9}+6$. For $m>5$

Proof. 

When $m=n$:

By using Table 1, we get,

$$\mathrm{ABC}\left(\mathcal{G}\right)={\displaystyle \sum _{uv\in E\left(\mathcal{G}\right)}}\sqrt{\frac{deg\left(u\right)+deg\left(v\right)-2}{deg\left(u\right)·deg\left(v\right)}}=\sum _{j=1}^3{\displaystyle \sum _{uv\in E_j\left(\mathcal{G}\right)}}\sqrt{\frac{deg\left(u\right)+deg\left(v\right)-2}{deg\left(u\right)·deg\left(v\right)}}.$$

$$\mathrm{ABC}\left(\mathcal{G}\right)=\frac{2}{3}|E_1\left(\mathcal{G}\right)|+\frac{1}{3}\sqrt{\frac{7}{2}}|E_2\left(\mathcal{G}\right)|+\frac{\sqrt{10}}{6}\left|E_3\left(\mathcal{G}\right)\right|$$

(9)

By doing some calculation, we get

$⟹\mathrm{ABC}\left(\mathcal{G}\right)$ = $\frac{1}{3}\sqrt{\frac{5}{2}}(3mn-6n+2)+\frac{1}{3}\sqrt{\frac{7}{2}}(3mn+3n-4)+\frac{2}{3}(3m+2)$

From Equation (7), we get:

$$\mathrm{GA}\left(\mathcal{G}\right)={\displaystyle \sum _{uv\in E\left(\mathcal{G}\right)}}\frac{2\sqrt{deg\left(u\right)deg\left(v\right)}}{\left(deg\right(u)+deg(v\left)\right)}=\sum _{j=1}^3{\displaystyle \sum _{uv\in E_j\left(\mathcal{G}\right)}}\frac{2\sqrt{deg\left(u\right)deg\left(v\right)}}{\left(deg\right(u)+deg(v\left)\right)}.$$

After some simplification, we get

$$\mathrm{GA}\left(\mathcal{G}\right)=\frac{2}{3}|E_1\left(\mathcal{G}\right)|+\frac{2}{3}\sqrt{2}|E_2\left(\mathcal{G}\right)|+|E_3\left(\mathcal{G}\right)|$$

(10)

By doing some calculation, we get:

$$\mathrm{GA}=6mn+3m-3n$$

(11)

For $S_u$ and $S_v$, the edge set $E\left(\mathcal{G}\right)$ can be divided into fourteen edge partitions. The $E_1\left(\mathcal{G}\right)$ contains six edges $uv$; where $S_u=S_v=12$. The $E_2\left(\mathcal{G}\right)$ have $3m-4$ edges $uv$, where $S_u=S_v=15$. The $E_3\left(\mathcal{G}\right)$ have $4m+4$ edges $uv$, where $S_u=15$ and $S_v=24$. The $E_4\left(\mathcal{G}\right)$ have six edges $uv$; where $S_u=12$ and $S_v=24$. The $E_5\left(\mathcal{G}\right)$ have $4m+4n-20$ edges $uv$, where we have $S_u=15$ and $S_v=27$. The $E_6\left(\mathcal{G}\right)$ have $2m$ edges $uv$, where $S_u=24$ and $S_v=27$. The $E_7\left(\mathcal{G}\right)$ have two edges $uv$, where $S_u=S_v=24$. The $E_8\left(\mathcal{G}\right)$ have $2m-8$ edges $uv$, where $S_u=S_v=27$. The $E_9\left(\mathcal{G}\right)$ have two edges $uv$, where $S_u=18$ and $S_v=24$. The $E_{10}\left(\mathcal{G}\right)$ have $2m+2n-10$ edges $uv$, where $S_u=18$ and $S_v=27$. The $E_{11}\left(\mathcal{G}\right)$ have $3mn-4m-9n+14$ edges $uv$, where $S_u=18$ and $S_v=30$. The $E_{12}\left(\mathcal{G}\right)$ have two edges $uv$, where $S_u=24$ and $S_v=30$. The $E_{13}\left(\mathcal{G}\right)$ have $3m+3n-14$ edges $uv$, where $S_u=27$ and $S_v=30$. The $E_{14}\left(\mathcal{G}\right)$ have $3mn-16m+20$ edges $uv$, where $S_u=S_v=30$. Table 2 shows such an edge partition of $\mathcal{G}$.

From Equation (6), we get:

$${\mathrm{ABC}}_4\left(\mathcal{G}\right)={\displaystyle \sum _{uv\in E\left(\mathcal{G}\right)}}\sqrt{\frac{S_u+S_v-2}{S_u{S}_v}}=\sum _{j=1}^{14}{\displaystyle \sum _{uv\in E_j\left(\mathcal{G}\right)}}\sqrt{\frac{S_u+S_v-2}{S_u{S}_v}}.$$

By using the edge partition given in

Table 10. Edge partition of silicate chain ($SL(m,n)$) based on the sum of degrees of the end vertices of each edge.

$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges	$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(12,12)$	6	$(27,27)$	$2m-8$
$(15,15)$	$3m-4$	$(18,24)$	2
$(15,24)$	$4m+4$	$(18,27)$	$2m+2n-10$
$(12,24)$	6	$(18,30)$	$3mn-4m-9n+14$
$(15,27)$	$4m+4n-20$	$(24,30)$	2
$(24,27)$	$2m$	$(27,30)$	$3m+3n-14$
$(24,24)$	2	$(30,30)$	$3mn-16m+20$

, we get the following:

${\mathrm{ABC}}_4\left(\mathcal{G}\right)$ = $\frac{\sqrt{22}}{12}|E_1\left(\mathcal{G}\right)|+\frac{2\sqrt{7}}{15}|E_2\left(\mathcal{G}\right)|+\frac{1}{6}\sqrt{\frac{37}{10}}|E_3\left(\mathcal{G}\right)|+\frac{1}{12}\sqrt{\frac{34}{2}}|E_4\left(\mathcal{G}\right)|+\frac{2\sqrt{10}}{\sqrt{405}}|E_5\left(\mathcal{G}\right)|+\frac{7}{18\sqrt{2}}|E_6\left(\mathcal{G}\right)|+\frac{\sqrt{46}}{24}|E_7\left(\mathcal{G}\right)|+\frac{2\sqrt{13}}{27}|E_8\left(\mathcal{G}\right)|+\frac{1}{6}\sqrt{\frac{10}{3}}|E_9\left(\mathcal{G}\right)|+\frac{1}{9}\sqrt{\frac{43}{6}}|E_{10}\left(\mathcal{G}\right)|+\frac{1}{6}\sqrt{\frac{46}{15}}|E_{11}\left(\mathcal{G}\right)|+\frac{1}{6}\sqrt{\frac{13}{5}}|E_{12}\left(\mathcal{G}\right)|+\frac{1}{9}\sqrt{\frac{55}{10}}|E_{13}\left(\mathcal{G}\right)|+\frac{7\sqrt{2}}{30}\left|E_{14}\left(\mathcal{G}\right)\right|$.

After some calculation, we get

$⟹{\mathrm{ABC}}_4\left(\mathcal{G}\right)$ = $\frac{1}{15}\sqrt{\frac{29}{2}}(3mn-16m+20)+\frac{2}{15}\sqrt{7}(3m-4)+\frac{1}{3}\sqrt{\frac{23}{30}}(3mn-9n-4m+14)+\frac{2}{27}\sqrt{13}(2m-8)+\frac{7m}{9\sqrt{2}}+\frac{1}{6}\sqrt{\frac{37}{10}}(4m+4)+\frac{1}{9}\sqrt{\frac{43}{6}}(2m+2n-10)+\frac{1}{9}\sqrt{\frac{11}{2}}(3m+3n-14)+\frac{2}{9}\sqrt{2}(4m+4n-20)+\frac{\sqrt{17}}{2}+\frac{\sqrt{\frac{23}{2}}}{6}+\frac{\sqrt{\frac{13}{5}}}{3}+\frac{\sqrt{\frac{10}{3}}}{3}$.

From Equation (8), we get:

$${\mathrm{GA}}_5\left(\mathcal{G}\right)={\displaystyle \sum _{uv\in E\left(\mathcal{G}\right)}}\frac{2\sqrt{S_u{S}_v}}{(S_u+S_v)}=\sum _{j=1}^{14}{\displaystyle \sum _{uv\in E_j\left(\mathcal{G}\right)}}\frac{2\sqrt{S_u{S}_v}}{(S_u+S_v)}$$

(12)

By using the edge partition given in Table 10, we have

${\mathrm{GA}}_5\left(\mathcal{G}\right)$ = $|E_1\left(\mathcal{G}\right)|+\frac{15}{12}|E_2\left(\mathcal{G}\right)|+\frac{4\sqrt{10}}{13}|E_3\left(\mathcal{G}\right)|+\frac{2\sqrt{2}}{3}|E_4\left(\mathcal{G}\right)|+\frac{3\sqrt{5}}{7}|E_5\left(\mathcal{G}\right)|+\frac{12\sqrt{2}}{17}|E_6\left(\mathcal{G}\right)|+|E_7\left(\mathcal{G}\right)|+|E_8\left(\mathcal{G}\right)|+\frac{4\sqrt{3}}{7}|E_9\left(\mathcal{G}\right)|+\frac{2\sqrt{6}}{5}|E_{10}\left(\mathcal{G}\right)|+\frac{\sqrt{15}}{4}|E_{11}\left(\mathcal{G}\right)|+\frac{4\sqrt{5}}{9}|E_{12}\left(\mathcal{G}\right)|+\frac{6\sqrt{10}}{19}|E_{13}|\left(\mathcal{G}\right)+|E_{14}\left(\mathcal{G}\right)|$.

After calculation, we get

$⟹{\mathrm{GA}}_5\left(\mathcal{G}\right)$ = $3mn+\frac{1}{4}\sqrt{15}(3mn-9n-4m+14)+\frac{24\sqrt{2}m}{17}+\frac{4}{13}\sqrt{10}(4m+4)+\frac{2}{5}\sqrt{6}(2m+2n-10)+\frac{6}{19}\sqrt{10}(3m+3n-14)+\frac{3}{7}\sqrt{5}(4m+4n-20)-11m+4\sqrt{2}+\frac{8\sqrt{3}}{7}+\frac{8\sqrt{5}}{9}+16$.

When $m<n$:

The proof of the following cases, (i) when m is odd and n is even, (ii) when m is even and n is odd, (iii) when m and n both are even, and (iv) when m and n both are odd, is the same as $m=n$ by using Table 2 and

Table 11. Edge partition of silicate chain ($SL(m,n)$), when $m<n$ and m is odd, based on the sum of the degrees of the end vertices of each edge.

$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges	$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(12,12)$	6	$(27,27)$	$m+n-7$
$(15,15)$	$3m-3$	$(18,24)$	2
$(15,24)$	$2m+2n+2$	$(18,27)$	$m+3n-11$
$(12,24)$	6	$(18,30)$	$3mn-6m-7n+16$
$(15,27)$	$m+7n-23$	$(24,30)$	2
$(24,27)$	$m+n-1$	$(27,30)$	$m+5n-17$
$(24,24)$	2	$(30,30)$	$3mn-4m-12n+26$

, Table 3 and

Table 12. Edge partition of silicate chain ($SL(m,n)$), when $m<n$ and m is even, based on the sum of the degrees of the end vertices of each edge.

$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges	$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(12,12)$	12	$(27,27)$	$m+n-7$
$(15,15)$	$3m-6$	$(18,24)$	4
$(15,24)$	$m+3n-11$	$(18,27)$	$m+3n-11$
$(12,24)$	12	$(18,30)$	$3mn-5m-8n+18$
$(15,27)$	$m+7n-23$	$(24,30)$	4
$(24,27)$	$m+n-1$	$(27,30)$	$m+5n-17$
$(24,24)$	0	$(30,30)$	$3mn-4m-12n+26$

, Table 4 and

Table 13. Edge partition of silicate chain ($SL(m,n)$), when $m<n$, m and n are even, based on the sum of the degrees of the end vertices of each edge.

$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges	$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(12,12)$	6	$(27,27)$	$m+n-6$
$(15,15)$	$3m-2$	$(18,24)$	2
$(15,24)$	$m+3n-2$	$(18,27)$	$4m-6$
$(12,24)$	6	$(18,30)$	$3mn+3m-16n+36$
$(15,27)$	$2m+6n-24$	$(24,30)$	2
$(24,27)$	$m+n-2$	$(27,30)$	$m+5n-20$
$(24,24)$	2	$(30,30)$	$3mn-16m+8$

, and Table 5 and

Table 14. Edge partition of silicate chain ($SL(m,n)$), when $m<n$, m and n are odd, based on the sum of the degrees of the end vertices of each edge.

$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges	$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(12,12)$	6	$(27,27)$	$m+n-6$
$(15,15)$	$3m-2$	$(18,24)$	2
$(15,24)$	$m+3n-2$	$(18,27)$	$m+3n-12$
$(12,24)$	6	$(18,30)$	$3mn-13m+4$
$(15,27)$	$m+7n-26$	$(24,30)$	2
$(24,27)$	$m+n-2$	$(27,30)$	$m+5n-20$
$(24,24)$	2	$(30,30)$	$3mn+4m-20n+48$

, respectively.

When: $m>n$:

The proof of the following cases, (i) when m is odd and n is even, (ii) when m is even and n is odd, (iii) when m and n both are even, and (iv) when m and n both are odd, is the same as $m=n$ by using Table 6 and

Table 15. Edge partition of silicate chain ($SL(m,n)$), when $m>n$ and m is odd, based on the sum of the degrees of the end vertices of each edge.

$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges	$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(12,12)$	6	$(27,27)$	$m+n-9$
$(15,15)$	$m+2n-3$	$(18,24)$	2
$(15,24)$	$m+3n+7$	$(18,27)$	$m+3n-9$
$(12,24)$	6	$(18,30)$	$3mn-5m-8n+11$
$(15,27)$	$m+7n-17$	$(24,30)$	2
$(24,27)$	$m+n+1$	$(27,30)$	$m+5n-11$
$(24,24)$	2	$(30,30)$	$3mn-2m-14n+12$

, Table 7 and

Table 16. Edge partition of silicate chain ($SL(m,n)$), when $m>n$ and m is even, based on the sum of the degrees of the end vertices of each edge.

$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges	$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(12,12)$	12	$(27,27)$	$m+n-9$
$(15,15)$	$m+2n-6$	$(18,24)$	4
$(15,24)$	$m+3n-5$	$(18,27)$	$m+3n-9$
$(12,24)$	12	$(18,30)$	$3mn-5m-8n+12$
$(15,27)$	$m+7n-17$	$(24,30)$	4
$(24,27)$	$m+n+1$	$(27,30)$	$m+5n-11$
$(24,24)$	0	$(30,30)$	$3mn-2m-14n+12$

, Table 8 and

Table 17. Edge partition of silicate chain ($SL(m,n)$), when $m>n$, m and n are even, based on the sum of the degrees of the end vertices of each edge.

$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges	$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(12,12)$	6	$(27,27)$	$m+n-10$
$(15,15)$	$3m-6$	$(18,24)$	2
$(15,24)$	$m+3n+10$	$(18,27)$	$m+3n-8$
$(12,24)$	6	$(18,30)$	$3mn-7m-6n+12$
$(15,27)$	$m+7n-14$	$(24,30)$	2
$(24,27)$	$m+n+2$	$(27,30)$	$m+5n-8$
$(24,24)$	2	$(30,30)$	$3mn-2m-14n+4$

, and Table 9 and

Table 18. Edge partition of silicate chain ($SL(m,n)$), when $m>n$, m and n are odd, based on the sum of the degrees of the end vertices of each edge.

$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges	$({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}})$ Where $\mathit{uv}\in \mathit{E}\left(\mathcal{G}\right)$	Number of Edges
$(12,12)$	6	$(27,27)$	$m+n-10$
$(15,15)$	$3n$	$(18,24)$	2
$(15,24)$	$m+3n+10$	$(18,27)$	$m+3n-8$
$(12,24)$	6	$(18,30)$	$3mn-2m-11n+2$
$(15,27)$	$m+7n-14$	$(24,30)$	2
$(24,27)$	$m+n+2$	$(27,30)$	$m+5n-8$
$(24,24)$	2	$(30,30)$	$3mn-4m-12n+8$

, respectively. □

3. Conclusions

In this paper, we have studied and computed some degree-based topological indices for the mth chain silicate $SL(m,n)$ network for the very first time. We have also calculated their structural closed formulas. These results help from the chemical point of view, as well as that of pharmaceutical science. Our research work has continued to derive new architectures from the chain silicate $SL(m,n)$ network.