Computing Topological Indices and Polynomials for Line Graphs

Abstract

A topological index is a number related to the atomic index that allows quantitative structure–action/property/toxicity connections. All the more vital topological indices correspond to certain physico-concoction properties like breaking point, solidness, strain vitality, and so forth, of synthetic mixes. The idea of the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials was set up in the substance diagram hypothesis in light of vertex degrees. These indices are valuable in the investigation of calming exercises of certain compound systems. In this paper, we computed the first and second Zagreb index, the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials of the line graph of wheel and ladder graphs by utilizing the idea of subdivision.

1. Introduction

Chemical graph theory is a branch of mathematical chemistry in which we apply apparatuses of the graph hypothesis to display the substance numerically. This hypothesis contributes noticeably to the synthetic sciences. A sub-atomic diagram is a straightforward limited graph in which vertices mean that the atoms and edges indicate concoction bonds in hidden compound structure. A topological index is actually a numerical amount related to the concoction constitution indicating the connection of the substance structure with numerous physio-synthetic properties, compound reactivity, and organic action. A decade ago, the diagram hypothesis found extensive use in research. The graph hypothesis has given physicists a variety of valuable apparatuses, such as topological files. Cheminformatics is a new subject that is a mix of science, arithmetic, and data science. It ponders quantitative structure–movement (QSAR) and structure–property (QSPR) connections that are utilized to anticipate the natural exercises and properties of synthetic mixes.

A graph G with vertex set V and edge set E are associated if there exists a connection between any combination of vertices in G. A network is just a connected diagram having no various edges and no self loops. For a graph G, the level of a vertex v is the quantity of edges occurrence to v and is indicated by $\xi \left(v\right)$.

A topological list $Top\left(G\right)$ of a graph G is a number with the property that for each chart H isomorphic to G, $Top\left(H\right)=Top\left(G\right)$. The idea of the topological file originated from the work done by Wiener [1], while at the same time, he was aiming to determine the breaking point of paraffin. He named this list as the way number. Later on, the way number was renamed as the Wiener index. The Wiener list is the first and most concentrated topological list, both from hypothetical and applications perspectives, and is characterized as the aggregate of separations between all sets of vertices in G (see [2] for details).

I. Gutman and N.Trinajstic [3] introduced the first and second Zagreb indices based on the degree of vertices as:

$$M_1\left(G\right)=\sum _{r_1{r}_2\in E\left(G\right)}[\xi \left(r_1\right)+\xi \left(r_2\right)],$$

(1)

$$M_2\left(G\right)=\sum _{r_1{r}_2\in E\left(G\right)}[\xi \left(r_1\right)\times \xi \left(r_2\right)].$$

(2)

In 2013, Shirdel et al. [4] introduced the “hyper Zagreb index” as:

$$HM\left(G\right)=\sum _{r_1{r}_2\in E\left(G\right)}{\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]}^2.$$

(3)

M. Ghorbani and N. Azimi defined [5] multiple Zagreb indices as:

$$P{M}_1\left(G\right)=\prod _{r_1{r}_2\in E\left(G\right)}[\xi \left(r_1\right)+\xi \left(r_2\right)],$$

(4)

$$P{M}_2\left(G\right)=\prod _{r_1{r}_2\in E\left(G\right)}[\xi \left(r_1\right)\times \xi \left(r_2\right)].$$

(5)

The properties of $P{M}_1\left(G\right),P{M}_2\left(G\right)$ indices for some chemical structures have been studied in [6]. The first Zagreb polynomial $M_1(G,x)$ ) and second Zagreb polynomial $M_2(G,x)$ are defined as:

$$M_1(G,x)=\sum _{r_1{r}_2\in E\left(G\right)}{x}^{[\xi \left(r_1\right)+\xi \left(r_2\right)]},$$

(6)

$$M_2(G,x)=\sum _{r_1{r}_2\in E\left(G\right)}{x}^{[\xi \left(r_1\right)\times \xi \left(r_2\right)]}.$$

(7)

There is now extensive research activity on $HM\left(G\right)$, $P{M}_1\left(G\right)$, $P{M}_2\left(G\right)$ indices and $M_1(G,x)$, $M_2(G,x)$ polynomials. See [7,8,9] for details.

2. Applications of Topological Indices

A ago, graph hypothesis had found an amazing use in research. Compound graph speculation has given researchers a variety of important gadgets (e.g., topological files). The Zagreb index is a topological descriptor that is related to a considerable measure of fabricated attributes of the particles, and has been discovered parallel to setting up the limit and Kovats constants of the particles [10]. The particle bond arranged hyper Zagreb index has a superior relationship with the security of direct dendrimers, besides the expanded medication stores and for setting up the strain criticalness of cycloalkanes [11,12,13,14,15]. To relate with certain physico-mix properties, different Zagreb indices have particularly needed insightful control over the farsighted essentialness of the dendrimers [16,17]. The first and second Zagreb polynomials were found to aid in the calculation of the aggregate $\pi$-electron imperativeness of the particles inside particular brutal verbalizations [18,19].

3. Topological Indices for Line Graph of Subdivided Graph $\mathit{L}\left(\mathit{S}\right(\mathit{G}\left)\right)$

The subdivision graph [20] $S\left(G\right)$ is the diagram acquired from G by supplanting every one of its edges by a way of length 2, or equivalently, by embedding an extra vertex into each edge of G. The line diagram of the chart G, composed $L\left(G\right)$, is the basic diagram whose vertices are the edges of G, with $ef\in E\left(L\right(G\left)\right)$ when e and f have a typical end point in G. Likewise, the line chart of the subdivided diagram is indicated by $L\left(S\right(G\left)\right)$.

The tadpole graph $T_{n,k}$ is the diagram acquired by joining a cycle diagram $C_n$ to a way of length k. By beginning with a disjoint association of two charts $G_1$ and $G_2$ and including edges joining every vertex of $G_1$ to that of $G_2$, one gets the whole $G_1+G_2$ of $G_1$ and $G_2$. The total $C_n+K_1$ of a cycle $C_n$ and a solitary vertex is alluded to as a wheel chart $W_{n+1}$ with arrange n. The Cartesian product $G_1\times G_2$ of charts $G_1$ and $G_2$ is a diagram with vertex set $V_1\times V_2$, and two vertices $r_1,s_1$ and $r_2,s_2$ are nearby in $G_1\times G_2$ if and only if either $r_1=r_2$ and $s_1{s}_2\in E_2$, or $s_1=s_2$ and $r_1{r}_2\in E_1$. The stepping stool diagram $L_n$ is given by $L_n=K_2\times P_n$, where $P_n$ is the way of length n. It is along these lines proportionate to the framework chart $G_{2,n}$. The diagram acquired by means of this definition resembles a stepping stool, having two rails and n rungs between them.

In 2011, Ranjini et al. figured the unequivocal articulations for the Schultz lists of the subdivision diagrams of the tadpole, wheel, steerage, and stepping stool charts. They additionally contemplated the Zagreb records of the line diagrams of the tadpole, haggle charts with subdivision in [21,22]. Ali et al. [23] registered the topological lists for the line diagram of the sparkler chart, and Sardar et al. [24] computed the topological files of the line diagrams of Banana tree and Firecracker diagrams. Ahmad et al. [25] discuss the m-polynomials and degree-based topological indices for the line graph of the Firecracker graph. Soleimani et al. [26] discuss the topological properties of nanostructures. In 2015, Su and Xu figured the general aggregate availability records and co-lists of the line diagrams of the tadpole and haggle charts with subdivision in [27]. Nadeem et al. [28,29] registered $AB{C}_4$ and $G{A}_5$ records of the line charts of the tadpole, wheel, stepping stool, $2D-$lattice, nanotube, and nanotorus of $TU{C}_4{C}_8[p,q]$ diagrams.

3.1. Zagreb Indices and Zagreb Polynomials of the Line Graph of the Tadpole Graph ${\mathit{T}}_{\mathit{n}\mathbf{,}\mathit{k}}$

Theorem 1.

Let R be the line graph of the tadpole graph $T_{n,k}$. Then

$$\begin{array}{ccc}\hfill M_1\left(R\right)& =& 8n+8k+12,\hfill \\ \hfill M_2\left(R\right)& =& 8n+8k+23,\hfill \\ \hfill HM\left(R\right)& =& 32n+32k+96,\hfill \\ \hfill P{M}_1\left(R\right)& =& 3\times 4^{(2n+2k-6)}\times 5^3\times 6^3,\hfill \\ \hfill P{M}_2\left(R\right)& =& 2\times 4^{(2n+2k-6)}\times 6^3\times 9^3,\hfill \\ \hfill M_1(R,x)& =& x^3+(2n+2k-6)x^4+3x^5+3x^6,\hfill \\ \hfill M_2(R,x)& =& x^2+(2n+2k-6)x^4+3x^6+3x^9.\hfill \end{array}$$

Proof. 

The subdivision diagram of $T_{n,k}$ and the related line chart R appear individually in Figure 1a,b. The subdivision chart $S\left(T_{n,k}\right)$ contains $2n+2k$ edges, so its line diagram contains $2n+2k$ vertices, out of which 3 vertices are of degree 3 and one vertex is of degree 1. The rest of the $2n+2k-4$ vertices are all of degree 2. It is easy to see that the aggregate number of edges of R is $2n+2k+1$. The edge set $E\left(R\right)$ separates into our edge segments in view of degrees of end vertices:

• $E_{12}\left(R\right)=\left\{r_1{r}_2\in E\left(R\right)\mid \xi \left(r_1\right)=1,\xi \left(r_2\right)=2\right\}$,
• $E_{22}\left(R\right)=\left\{r_1{r}_2\in E\left(R\right)\mid \xi \left(r_1\right)=2,\xi \left(r_2\right)=2\right\}$,
• $E_{23}\left(R\right)=\left\{r_1{r}_2\in E\left(R\right)\mid \xi \left(r_1\right)=2,\xi \left(r_2\right)=3\right\}$,
• $E_{33}\left(R\right)=\left\{r_1{r}_2\in E\left(R\right)\mid \xi \left(r_1\right)=3,\xi \left(r_2\right)=3\right\}$.

 ☐

These four partitions of the edge set correspond to their degree sum of neighbors of end vertices. The number of edges in $E_{12}\left(R\right)$ is 1, in $E_{22}\left(R\right)$ there are $2n+2k-6$, in $E_{23}\left(R\right)$ there are 3, and in $E_{33}\left(R\right)$ there are 3. Now, using Equations (1)–(7), we have

$$\begin{array}{ccc}\hfill M_1\left(G\right)& =& \sum _{r_1{r}_2\in E\left(G\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right],\hfill \\ \hfill M_1\left(R\right)& =& \sum _{r_1{r}_2\in E_{12}\left(R\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]+\sum _{r_1{r}_2\in E_{22}\left(R\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]+\sum _{r_1{r}_2\in E_{23}\left(R\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]\hfill \\ & +& \sum _{r_1{r}_2\in E_{33}\left(R\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]\hfill \\ & =& 3|E_{12}\left(R\right)|+4|E_{22}\left(R\right)|+5|E_{23}\left(R\right)|+6|E_{33}\left(R\right)|\hfill \\ & =& 3\left(1\right)+4\left(\right(2n+2k-6\left)\right)+5\left(3\right)+6\left(3\right)=8n+8k+12,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill M_2\left(G\right)& =& \sum _{r_1{r}_2\in E\left(G\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\hfill \\ \hfill M_2\left(R\right)& =& \sum _{r_1{r}_2\in E_{12}\left(R\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]+\sum _{r_1{r}_2\in E_{22}\left(R\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]+\sum _{r_1{r}_2\in E_{23}\left(R\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\hfill \\ & +& \sum _{r_1{r}_2\in E_{33}\left(R\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\hfill \\ & =& 2|E_{12}\left(R\right)|+4|E_{22}\left(R\right)|+6|E_{23}\left(R\right)|+9|E_{33}\left(R\right)|\hfill \\ & =& 2\left(1\right)+4\left(\right(2n+2k-6\left)\right)+6\left(3\right)+9\left(3\right)=8n+8k+23,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill HM\left(G\right)& =& \sum _{r_1{r}_2\in E\left(G\right)}{\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]}^2,\hfill \\ \hfill HM\left(R\right)& =& \sum _{r_1{r}_2\in E_{12}\left(R\right)}{\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]}^2+\sum _{r_1{r}_2\in E_{22}\left(R\right)}{\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]}^2+\sum _{r_1{r}_2\in E_{23}\left(R\right)}{\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]}^2\hfill \\ & +& \sum _{r_1{r}_2\in E_{33}\left(R\right)}{\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]}^2\hfill \\ & =& 9|E_{12}\left(R\right)|+16|E_{22}\left(R\right)|+25|E_{23}\left(R\right)|+36|E_{33}\left(R\right)|\hfill \\ & =& 9\left(1\right)+16(2n+2k-6)+25\left(3\right)+36\left(3\right)=2n+32k+96,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill P{M}_1\left(G\right)& =& \prod _{r_1{r}_2\in E\left(G\right)}[\xi \left(r_1\right)+\xi \left(r_2\right)],\hfill \\ \hfill P{M}_1\left(R\right)& =& \prod _{r_1{r}_2\in E_{12}\left(R\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]\times \prod _{r_1{r}_2\in E_{22}\left(R\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]\times \prod _{r_1{r}_2\in E_{23}\left(R\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]\hfill \\ & \times & \prod _{r_1{r}_2\in E_{33}\left(R\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]\hfill \\ & =& 3^{|E_{12}\left(R\right)|}\times 4^{|E_{22}\left(R\right)|}\times 5^{|E_{23}\left(R\right)|}\times 6^{|E_{33}\left(R\right)|}=3\times 4^{(2n+2k-6)}\times 5^3\times 6^3,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill P{M}_2\left(G\right)& =& \prod _{r_1{r}_2\in E\left(G\right)}[\xi \left(r_1\right)\times \xi \left(r_2\right)],\hfill \\ \hfill P{M}_2\left(R\right)& =& \prod _{r_1{r}_2\in E_{12}\left(R\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\times \prod _{r_1{r}_2\in E_{22}\left(R\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\times \prod _{r_1{r}_2\in E_{23}\left(R\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\hfill \\ & \times & \prod _{r_1{r}_2\in E_{33}\left(R\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\hfill \\ & =& 2^{|E_{12}\left(R\right)|}\times 4^{|E_{22}\left(R\right)|}\times 6^{|E_{23}\left(R\right)|}\times 9^{|E_{33}\left(R\right)|}=2\times 4^{(2n+2k-6)}\times 6^3\times 9^3,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill M_1(G,x)& =& \sum _{r_1{r}_2\in E\left(G\right)}{x}^{[\xi \left(r_1\right)+\xi \left(r_2\right)]},\hfill \\ \hfill M_1(R,x)& =& \sum _{r_1{r}_2\in E_1\left(R\right)}{x}^{[\xi \left(r_1\right)+\xi \left(r_2\right)]}+\sum _{r_1{r}_2\in E_2\left(R\right)}{x}^{[\xi \left(r_1\right)+\xi \left(r_2\right)]}+\sum _{r_1{r}_2\in E_3\left(R\right)}{x}^{[\xi \left(r_1\right)+\xi \left(r_2\right)]}\hfill \\ & +& \sum _{r_1{r}_2\in E_4\left(R\right)}{x}^{[\xi \left(r_1\right)+\xi \left(r_2\right)]}\hfill \\ & =& \sum _{r_1{r}_2\in E_1\left(R\right)}{x}^3+\sum _{r_1{r}_2\in E_2\left(R\right)}{x}^4+\sum _{r_1{r}_2\in E_3\left(R\right)}{x}^5+\sum _{r_1{r}_2\in E_4\left(R\right)}{x}^6\hfill \\ & =& |E_{12}\left(R\right)|x^3+|E_{22}\left(R\right)|x^4+|E_{23}\left(R\right)|x^5+|E_{33}\left(R\right)|x^6=x^3+(2n+2k-6)x^4+3x^5+3x^6,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill M_2(G,x)& =& \sum _{r_1{r}_2\in E\left(G\right)}{x}^{[\xi \left(r_1\right)\times \xi \left(r_2\right)]},\hfill \\ \hfill M_2(R,x)& =& \sum _{r_1{r}_2\in E_1\left(R\right)}{x}^{[\xi \left(r_1\right)\times \xi \left(r_2\right)]}+\sum _{r_1{r}_2\in E_2\left(R\right)}{x}^{[\xi \left(r_1\right)\times \xi \left(r_2\right)]}+\sum _{r_1{r}_2\in E_3\left(R\right)}{x}^{[\xi \left(r_1\right)\times \xi \left(r_2\right)]}\hfill \\ & +& \sum _{r_1{r}_2\in E_4\left(R\right)}{x}^{[\xi \left(r_1\right)\times \xi \left(r_2\right)]}\hfill \\ & =& \sum _{r_1{r}_2\in E_1\left(R\right)}{x}^2+\sum _{r_1{r}_2\in E_2\left(R\right)}{x}^4+\sum _{r_1{r}_2\in E_3\left(R\right)}{x}^6+\sum _{r_1{r}_2\in E_4\left(R\right)}{x}^9\hfill \\ & =& |E_{12}\left(R\right)|x^2+|E_{22}\left(R\right)|x^4+|E_{23}\left(R\right)|x^6+|E_{33}\left(R\right)|x^9=x^2+(2n+2k-6)x^4+3x^6+3x^9.\hfill \end{array}$$

Theorem 2.

Let H be the line graph of the wheel graph $W_{n+1}$. Then

$$\begin{array}{ccc}\hfill M_1\left(H\right)& =& n^3+27n,\hfill \\ \hfill M_2\left(H\right)& =& \frac{n^4-n^3+6n^2+72n}{2},\hfill \\ \hfill HM\left(H\right)& =& 2n^4-n^3+6n^2+45n,\hfill \\ \hfill P{M}_1\left(H\right)& =& 6^{4n}\times {(3+n)}^n\times {\left(2n\right)}^{\left(\frac{n^2-n}{2}\right)},\hfill \\ \hfill P{M}_2\left(H\right)& =& 9^{4n}\times {\left(3n\right)}^n\times n^{n^2-n},\hfill \\ \hfill M_1(H,x)& =& 4n{x}^4+n{x}^{3+n}+\left(\frac{n^2-n}{2}\right)x^{2n},\hfill \\ \hfill M_2(H,x)& =& 4n{x}^9+n{x}^{3n}+\left(\frac{n^2-n}{2}\right)x^{n^2}.\hfill \end{array}$$

Proof. 

The subdivision chart of wheel $W_{n+1}$ and the relating line diagram H appear separately in Figure 2a,b. The line chart H contains $4n$ vertices, of which $3n$ vertices are of degree 3 and the others are of degree n. It is simple to determine that the aggregate number of edges in the line diagram H are ${\displaystyle \frac{n^2+9n}{2}}$. To demonstrate the above proclamation, the edge set $E\left(H\right)$ isolates into three edge segments in light of the degrees of end vertices:

• $E_{33}\left(H\right)=\left\{r_1{r}_2\in E\left(H\right)\mid \xi \left(r_1\right)=3,\xi \left(r_2\right)=3\right\}$,
• $E_{3n}\left(H\right)=\left\{r_1{r}_2\in E\left(H\right)\mid \xi \left(r_1\right)=3,\xi \left(r_2\right)=n\right\}$,
• $E_{nn}\left(H\right)=\left\{r_1{r}_2\in E\left(H\right)\mid \xi \left(r_1\right)=n,\xi \left(r_2\right)=n\right\}$.

 ☐

These three partitions of the edge set correspond to their degree sum of neighbors of end vertices. The number of edges in $E_{33}\left(H\right)$ are $4n$, in $E_{3n}\left(HG\right)$ there are n, and in $E_{nn}\left(H\right)$ there are ${\displaystyle \frac{n^2-n}{2}}$. Now, using Equations (1)–(7), we have:

$$\begin{array}{ccc}\hfill M_1\left(G\right)& =& \sum _{r_1{r}_2\in E\left(G\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right],\hfill \\ \hfill M_1\left(H\right)& =& \sum _{r_1{r}_2\in E_{33}\left(H\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]+\sum _{r_1{r}_2\in E_{3n}\left(H\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]+\sum _{r_1{r}_2\in E_{nn}\left(H\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]\hfill \\ & =& 6|E_{33}\left(H\right)|+(3+n)|E_{3n}\left(H\right)|+2n|E_{nn}\left(H\right)|\hfill \\ & =& 6\left(4n\right)+(3+n)\left(n\right)+2n(\frac{n^2-n}{2})=n^3+27n,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill M_2\left(G\right)& =& \sum _{r_1{r}_2\in E\left(G\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right],\hfill \\ \hfill M_2\left(H\right)& =& \sum _{r_1{r}_2\in E_{33}\left(H\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]+\sum _{r_1{r}_2\in E_{3n}\left(H\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]+\sum _{r_1{r}_2\in E_{nn}\left(H\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\hfill \\ & =& 9|E_{33}\left(H\right)|+3n|E_{3n}\left(H\right)|+n^2|E_{nn}\left(H\right)|\hfill \\ & =& 9\left(4n\right)+3n\left(n\right)+n^2(\frac{n^2-n}{2})=\frac{n^4-n^3+6n^2+72n}{2},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill HM\left(G\right)& =& \sum _{r_1{r}_2\in E\left(G\right)}{\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]}^2,\hfill \\ \hfill HM\left(H\right)& =& \sum _{r_1{r}_2\in E_{33}\left(H\right)}{\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]}^2+\sum _{r_1{r}_2\in E_{3n}\left(H\right)}{\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]}^2+\sum _{r_1{r}_2\in E_{nn}\left(H\right)}{\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]}^2\hfill \\ & =& 9|E_{33}\left(H\right)|+{(3+n)}^2|E_{3n}\left(H\right)|+4n^2|E_{nn}\left(H\right)|\hfill \\ & =& 9\left(4n\right)+n{(3+n)}^2+4n^2(\frac{n^2-n}{2})=2n^4-n^3+6n^2+45n,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill P{M}_1\left(G\right)& =& \prod _{r_1{r}_2\in E\left(G\right)}[\xi \left(r_1\right)+\xi \left(r_2\right)],\hfill \\ \hfill P{M}_1\left(H\right)& =& \prod _{r_1{r}_2\in E_{33}\left(H\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]\times \prod _{r_1{r}_2\in E_{3n}\left(H\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]\times \prod _{r_1{r}_2\in E_{nn}\left(H\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]\hfill \\ & =& 6^{|E_{33}\left(H\right)|}\times {(3+n)}^{|E_{3n}\left(H\right)|}\times 2n^{|E_3\left(H\right)|}\hfill \\ & =& 6^{4n}\times {(3+n)}^n\times {\left(2n\right)}^{(\frac{n^2-n}{2})},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill P{M}_2\left(G\right)& =& \prod _{r_1{r}_2\in E\left(G\right)}[\xi \left(r_1\right)\times \xi \left(r_2\right)],\hfill \\ \hfill P{M}_2\left(H\right)& =& \prod _{r_1{r}_2\in E_{33}\left(H\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\times \prod _{r_1{r}_2\in E_{3n}\left(H\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\times \prod _{r_1{r}_2\in E_{nn}\left(H\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\hfill \\ & =& 9^{|E_{33}\left(H\right)|}\times {\left(3n\right)}^{|E_{3n}\left(H\right)|}\times {\left(n^2\right)}^{|E_{nn}\left(H\right)|}\hfill \\ & =& 9^{4n}\times {\left(3n\right)}^n\times n^{n^2-n},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill M_1(G,x)& =& \sum _{r_1{r}_2\in E\left(G\right)}{x}^{[\xi \left(r_1\right)+\xi \left(r_2\right)]},\hfill \\ \hfill M_1(H,x)& =& \sum _{r_1{r}_2\in E_1\left(H\right)}{x}^{[\xi \left(r_1\right)+\xi \left(r_2\right)]}+\sum _{r_1{r}_2\in E_{3n}\left(H\right)}{x}^{[\xi \left(r_1\right)+\xi \left(r_2\right)]}+\sum _{r_1{r}_2\in E_{nn}\left(H\right)}{x}^{[\xi \left(r_1\right)+\xi \left(r_2\right)]}\hfill \\ & =& \sum _{r_1{r}_2\in E_1\left(H\right)}{x}^6+\sum _{r_1{r}_2\in E_2\left(H\right)}{x}^{3+n}+\sum _{r_1{r}_2\in E_3\left(H\right)}{x}^{2n}\hfill \\ & =& |E_{33}\left(H\right)|x^6+|E_{3n}\left(H\right)|x^{3+n}+|E_{nn}\left(H\right)|x^{2n}\hfill \\ & =& 4n{x}^4+n{x}^{3+n}+(\frac{n^2-n}{2})x^{2n},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill M_2(G,x)& =& \sum _{r_1{r}_2\in E\left(G\right)}{x}^{[\xi \left(r_1\right)\times \xi \left(r_2\right)]},\hfill \\ \hfill M_2(H,x)& =& \sum _{r_1{r}_2\in E_1\left(H\right)}{x}^{[\xi \left(r_1\right)\times \xi \left(r_2\right)]}+\sum _{r_1{r}_2\in E_{3n}\left(H\right)}{x}^{[\xi \left(r_1\right)\times \xi \left(r_2\right)]}+\sum _{r_1{r}_2\in E_{nn}\left(H\right)}{x}^{[\xi \left(r_1\right)\times \xi \left(r_2\right)]}\hfill \\ & =& \sum _{r_1{r}_2\in E_1\left(H\right)}{x}^9+\sum _{r_1{r}_2\in E_2\left(H\right)}{x}^{3n}+\sum _{r_1{r}_2\in E_3\left(H\right)}{x}^{n^2}\hfill \\ & =& |E_{33}\left(H\right)|x^9+|E_{3n}\left(H\right)|x^{3n}++|E_{nn}\left(H\right)|x^{n^2}\hfill \\ & =& 4n{x}^9+n{x}^{3n}+(\frac{n^2-n}{2})x^{n^2}.\hfill \end{array}$$

Theorem 3.

Let $P_n$ be the line graph of the ladder graph $L_n$ of order n. Then,

$$\begin{array}{ccc}\hfill M_1\left(P_n\right)& =& 154n-76,\hfill \\ \hfill M_2\left(P_n\right)& =& 81n-132,\hfill \\ \hfill HM\left(P_n\right)& =& 324n-524,\hfill \\ \hfill P{M}_1\left(P_n\right)& =& 4^6\times 5^4\times 6^{(9n-20)},\hfill \\ \hfill P{M}_2\left(P_n\right)& =& 4^6\times 6^4\times 9^{(9n-20)},\hfill \\ \hfill M_1(P_n,x)& =& 6x^4+4x^5+(9n-20)x^6,\hfill \\ \hfill M_2(P_n,x)& =& 6x^4+4x^6+(9n-20)x^9.\hfill \end{array}$$

Proof. 

The subdivision diagram of the stepping stool chart $L_n$ and the comparing line chart $P_n$ appear in Figure 3a,b, respectively. The quantity of vertices in the line chart $P_n$ are $6n-4$, among which 8 vertices are of degree 2 and the rest of the $6n-12$ vertices are of degree 3. It is simple to process that the aggregate number of edges in the line chart $P_n$ is $9n-10$. To demonstrate the above proclamation, the edge set $E\left(P_n\right)$ isolates into three edge parcels in light of the degrees of end vertices:

• $E_{22}\left(P_n\right)=\left\{r_1{r}_2\in E\left(P_n\right)\mid \xi \left(r_1\right)=2,\xi \left(r_2\right)=2\right\}$,
• $E_{23}\left(P_n\right)=\left\{r_1{r}_2\in E\left(P_n\right)\mid \xi \left(r_1\right)=2,\xi \left(r_2\right)=3\right\}$,
• $E_{33}\left(P_n\right)=\left\{r_1{r}_2\in E\left(P_n\right)\mid \xi \left(r_1\right)=3,\xi \left(r_2\right)=3\right\}$.

 ☐

These three partitions of the edge set correspond to their degree sum of neighbors of end vertices. The number of edges in $E_{22}\left(P_n\right)$ are 6, in $E_{23}\left(P_n\right)$ there are 4, and in $E_{33}\left(P_n\right)$ there are $9n-20$. Now, using Equations (1)–(7), we have:

$$\begin{array}{ccc}\hfill M_1\left(G\right)& =& \sum _{r_1{r}_2\in E\left(G\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right],\hfill \\ \hfill M_1\left(P_n\right)& =& \sum _{r_1{r}_2\in E_{22}\left(P_n\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]+\sum _{r_1{r}_2\in E_{23}\left(P_n\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]+\sum _{r_1{r}_2\in E_{33}\left(P_n\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]\hfill \\ & =& 4|E_{22}\left(P_n\right)|+5|E_{23}\left(P_n\right)|+6|E_{33}\left(P_n\right)|\hfill \\ & =& 4\left(6\right)+5\left(4\right)+6(9n-20)=154n-76,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill M_2\left(G\right)& =& \sum _{r_1{r}_2\in E\left(G\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right],\hfill \\ \hfill M_2\left(P_n\right)& =& \sum _{r_1{r}_2\in E_{22}\left(P_n\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]+\sum _{r_1{r}_2\in E_{23}\left(P_n\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]+\sum _{r_1{r}_2\in E_{33}\left(P_n\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\hfill \\ & =& 4|E_{22}\left(P_n\right)|+6|E_{23}\left(P_n\right)|+9|E_{33}\left(P_n\right)|\hfill \\ & =& 4\left(6\right)+6\left(4\right)+9(9n-20)=81n-132,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill HM\left(G\right)& =& \sum _{r_1{r}_2\in E\left(G\right)}{\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]}^2,\hfill \\ \hfill HM\left(P_n\right)& =& \sum _{r_1{r}_2\in E_{22}\left(P_n\right)}{\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]}^2+\sum _{r_1{r}_2\in E_{23}\left(P_n\right)}{\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]}^2+\sum _{r_1{r}_2\in E_{33}\left(P_n\right)}{\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]}^2\hfill \\ & =& 16|E_{22}\left(P_n\right)|+25|E_{23}\left(P_n\right)|+36|E_{33}\left(P_n\right)|\hfill \\ & =& 16\left(6\right)+25\left(4\right)+36(9n-20)=324n-524,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill P{M}_1\left(G\right)& =& \prod _{r_1{r}_2\in E\left(G\right)}[\xi \left(r_1\right)+\xi \left(r_2\right)],\hfill \\ \hfill P{M}_1\left(P_n\right)& =& \prod _{r_1{r}_2\in E_{22}\left(P_n\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]\times \prod _{r_1{r}_2\in E_{23}\left(P_n\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]\times \prod _{r_1{r}_2\in E_{33}\left(P_n\right)}\left[\xi \left(r_1\right)+\xi \left(r_2\right)\right]\hfill \\ & =& 4^{|E_{22}\left(P_n\right)|}\times 5^{|E_{23}\left(P_n\right)|}\times 6^{|E_{33}\left(P_n\right)|}=4^6\times 5^4\times 6^{(9n-20)},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill P{M}_2\left(G\right)& =& \prod _{r_1{r}_2\in E\left(G\right)}[\xi \left(r_1\right)\times \xi \left(r_2\right)],\hfill \\ \hfill P{M}_2\left(P_n\right)& =& \prod _{r_1{r}_2\in E_{22}\left(P_n\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\times \prod _{r_1{r}_2\in E_{23}\left(P_n\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\times \prod _{r_1{r}_2\in E_{33}\left(P_n\right)}\left[\xi \left(r_1\right)\times \xi \left(r_2\right)\right]\hfill \\ & =& 4^{|E_{22}\left(P_n\right)|}\times 6^{|E_{23}\left(P_n\right)|}\times 9^{|E_{33}\left(P_n\right)|}=4^6\times 6^4\times 9^{(9n-20)},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill M_1(G,x)& =& \sum _{r_1{r}_2\in E\left(G\right)}{x}^{[\xi \left(r_1\right)+\xi \left(r_2\right)]},\hfill \\ \hfill M_1(P_n,x)& =& \sum _{r_1{r}_2\in E_1\left(P_n\right)}{x}^{[\xi \left(r_1\right)+\xi \left(r_2\right)]}+\sum _{r_1{r}_2\in E_2\left(P_n\right)}{x}^{[\xi \left(r_1\right)+\xi \left(r_2\right)]}+\sum _{r_1{r}_2\in E_3\left(P_n\right)}{x}^{[\xi \left(r_1\right)+\xi \left(r_2\right)]}\hfill \\ & =& \sum _{r_1{r}_2\in E_1\left(P_n\right)}{x}^4+\sum _{r_1{r}_2\in E_2\left(P_n\right)}{x}^5+\sum _{r_1{r}_2\in E_3\left(P_n\right)}{x}^6\hfill \\ & =& |E_{22}\left(P_n\right)|x^4+|E_{23}\left(P_n\right)|x^5++|E_{33}\left(P_n\right)|x^6=6x^4+4x^5+(9n-20)x^6,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill M_2(G,x)& =& \sum _{r_1{r}_2\in E\left(G\right)}{x}^{[\xi \left(r_1\right)\times \xi \left(r_2\right)]},\hfill \\ \hfill M_2(P_n,x)& =& \sum _{r_1{r}_2\in E_1\left(P_n\right)}{x}^{[\xi \left(r_1\right)\times \xi \left(r_2\right)]}+\sum _{r_1{r}_2\in E_2\left(P_n\right)}{x}^{[\xi \left(r_1\right)\times \xi \left(r_2\right)]}+\sum _{r_1{r}_2\in E_3\left(P_n\right)}{x}^{[\xi \left(r_1\right)\times \xi \left(r_2\right)]}\hfill \\ & =& \sum _{r_1{r}_2\in E_1\left(P_n\right)}{x}^4+\sum _{r_1{r}_2\in E_2\left(P_n\right)}{x}^6+\sum _{r_1{r}_2\in E_3\left(P_n\right)}{x}^9\hfill \\ & =& |E_{22}\left(P_n\right)|x^4+|E_{23}\left(P_n\right)|x^6++|E_{33}\left(P_n\right)|x^9=6x^4+4x^6+(9n-20)x^9.\hfill \end{array}$$

4. Conclusions

In this paper we determined first and second Zagreb record, Hyper Zagreb index, first numerous Zagreb index, second different Zagreb index, and Zagreb polynomials of the line chart of tadpole and haggle diagrams by utilizing the idea of subdivision.

In the past couple of decades, investigations of the topological indices in view of end-vertex degrees of edges have seen a significant increase. The issue of determining the estimations of some outstanding degree-based topological indices is completely addressed for the line diagram of the subdivision graphs. This provides a path forward in this field of research. Also, in future we are intrigued to register these records for the line diagrams of some outstanding graphs.