## 1 Introduction  
All graphs discussed in this paper are simple and connected. Let \(G\) be a (molecular) graph with vertex set \(V(G)\) and edge set \(E(G)\), respectively. For a vertex \(u\in V(G)\), we denote \(N_{u}=\{v:uv\in E(G)\}\), \(d_{u}=|N_{u}|\) (degree of \(u\)) and \(S_{u}=\sum_{v\in N_{u}}d_{v}\). The subdivision of an edge \(uv\) in a graph \(G\) is deduced by inserting a new vertex \(w\) in \(V(G)\), and in edge set \(E(G)\) the edge \(uv\) is replaced by two new edges \(uw\) and \(wv\). In this way, the subdivision graph \(S(G)\) results from the subdivision of all the edges of \(G\). The line graph \(L(G)\) of a graph \(G\) is the graph with \(V(L(E))=E(G)\), and \(e_{1}e_{2}\in E(L(G))\) if and only if \(e_{1}\) and \(e_{2}\) share a common endpoint in \(G\). The \(L(S(G))\) is the line graph of \(S(G)\).  
The first degree-based topological index dates back to 1975, which is called the Randic index. The general Randic connectivity index [1] of graph \(G\) can be formulated by  
\[R_{\alpha}(G)=\sum_{uv\in E(G)}(d_{u}d_{v})^{\alpha}, \tag{1}\]  
where \(\alpha\in R\) (in what follows, \(\alpha\) always denoted as a real number), and \(R_{-1/2}(G)\) is well known as Randic connectivity index of graph \(G\).  
[PERSON] and [PERSON] [2] introduced the first general Zagreb index which is stated as  
\[M_{\alpha}(G)=\sum_{u\in V(G)}(d_{u})^{\alpha}. \tag{2}\]The general version of sum-connectivity index \(\chi_{\alpha}(G)\) has been introduced in 2010 [3], which is defined as  
\[\chi_{\alpha}(G)=\sum_{uv\in E(G)}(d_{u}+d_{v})^{\alpha}. \tag{3}\]  
[PERSON] et al. [4] defined the atom-bond connectivity (\(ABC\)) index which was widely studied by researchers in the recent 10 years. For a graph \(G\), its \(ABC\) index is expressed by  
\[ABC(G)=\sum_{uv\in E(G)}\sqrt{\frac{d_{u}+d_{v}-2}{d_{u}d_{v}}}. \tag{4}\]  
The fourth \(ABC\) index (denoted by \(ABC_{4}\), as a neighborhood degree-based analog invariant) was introduced by [PERSON] et al. in [5], which is stated as  
\[ABC_{4}(G)=\sum_{uv\in E(G)}\sqrt{\frac{S_{u}+S_{v}-2}{S_{u}S_{v}}}. \tag{5}\]  
[PERSON] and [PERSON] [6] introduced the geometric arithmetic (GA) index and defined it as  
\[GA(G)=\sum_{uv\in E(G)}\frac{2\sqrt{d_{u}d_{v}}}{d_{u}+d_{v}}. \tag{6}\]  
A variant of GA index, the so-call the fifth \(GA\) index \((GA_{5})\), is proposed by [PERSON] et al. [7] as  
\[GA_{5}(G)=\sum_{uv\in E(G)}\frac{2\sqrt{S_{u}S_{v}}}{S_{u}+S_{v}}. \tag{7}\]  
For more recent contributions on the topological indices of graphs in different settings the readers may refer to [8; 9; 10].  
[PERSON] et al. [21] presented the exact values of the Shultz index of the subdivision graph of the wheel, tadpole, ladder, and helm graphs. The Zagreb indices of the line graph with subdivision of these structure are studied in [17]. In light of graph structure analysis, [PERSON] and [PERSON] [22] computed the general sum-connectivity index and co-index of the line graph of these graphs with subdivision. [PERSON] et al. [23] determined \(ABC_{4}\) and \(GA_{5}\) indices of the line graph of the wheel, ladder, and tadpole graphs by means of the definition of subdivision.  
The strategies we used in our computations are: combinatorial registering, vertex partition procedure, edge partition technique, and degree-counting schemes. Furthermore, with the help of vertex and edge partitions, we compute various degree-based indices. In addition, we use Maple for calculations.  
This paper is structured as follows. In next section, we yield a close formula of general Randic index, general sum connectivity index, general zagreb index, \(ABC\), \(ABC_{4}\), \(GA\), and \(GA_{5}\) indices for \(L(S(G))\), where \(G\) is certain rooted product graph. We define the \(i\)-th vertex rooted product of graphs and study the topological indices for \(L(S(G))\), where \(G\) is a family of \(i\)-th vertex rooted product of graphs in Section 3. Some remarks are presented in the concluding section.