Computing Certain Topological Indices of the Line Graphs of Subdivision Graphs of Some Rooted Product Graphs

In this work, we study the degree-based topological invariants, and the general sum-connectivity, A B C 4 , G A 5 , general Zagreb, G A , generalized Randić, and A B C indices of the line graphs of some rooted product graphs ( C n { P k } and C n { S m + 1 } ) are determined by menas of the concept of subdivision. Moreover, we also computed all these indices of the line graphs of the subdivision graphs of i-th vertex rooted product graph C i , r { P k + 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 ∈ V(G), we denote N u = {v : uv ∈ E(G)}, d u = |N u | (degree of u) and S u = ∑ v∈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 ∈ 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 Randić index. The general Randić connectivity index [1] of graph G can be formulated by where α ∈ R (in what follows, α always denoted as a real number), and R −1/2 (G) is well known as Randić connectivity index of graph G. Li and Zhao [2] introduced the first general Zagreb index which is stated as The general version of sum-connectivity index χ α (G) has been introduced in 2010 [3], which is defined as Estrada 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 The fourth ABC index (denoted by ABC 4 , as a neighborhood degree-based analog invariant) was introduced by Ghorbani et al. in [5], which is stated as Vukicevic and Furtula [6] introduced the geometric arithmetic (GA) index and defined it as A variant of GA index, the so-call the fifth GA index (GA 5 ), is proposed by Graovac et al. [7] as For more recent contributions on the topological indices of graphs in different settings the readers may refer to [8][9][10][11][12][13][14][15][16][17][18][19][20].
Ranjini 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, Su and Xu [22] computed the general sum-connectivity index and co-index of the line graph of these graphs with subdivision. Nadeem 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 Randić 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.

Topological Indices of Line Graph of the Subdivision Graph of Rooted Product of Graphs
Let H be a labeled graph with vertex set V(H) = {1, 2, . . . , n} and G be a sequence of n rooted graphs G 1 , G 2 , . . . , G n . Godsil and Mckay [24] defined the rooted product of H by G, i.e., H(G) = H(G 1 , G 2 , . . . , G n ) is a graph yielded by identifying the root vertex of G i with the i-th vertex of H for all i ∈ {1, 2, . . . , n}. In particular, if the components G i (i ∈ {1, 2, . . . , n}) are mutually isomorphic to K, then the rooted product of H by G is denoted by H{K}, which is the so-called cluster product of H by K.

Topological Indices of Line Graph of the Subdivision Graph of C n {P k }
Let P n and C n denotes the path and cycle with order n. Let C n {P k+1 } be the cluster product of cycle of length n with a path of length k. The graph of C n {P k+1 } is shown in Figure 1. It clearly shows that the number of vertices and number of edges in C n {P k+1 } are both n + nk. In the next theorem we will compute the general Zagreb index of the line graph of subdivision graph of C n {P k+1 }.
Proof. The structure of G is shown in Figure 1. There are total 2n + 2nk vertices among which the number of vertices of degree 1, 2, and 3 are n, 2nk − 2n, and 3n, respectively. Thus, M α (G) is obtained in terms of Equation (2). Theorem 2. Let G be a graph isomorphic to the line graph of the subdivision graph of C n {P k+1 }. Then Proof. It is easily seen that |E(G)| = 3n + 2kn. The edge partitions, based on the degrees of the vertices for the cases k = 1 and k > 1, are shown in Tables 1 and 2, respectively. In view of (1), (3), (4), and (6), we infer the required conclusions.
Number of edges n 4n Table 2. The edge partition of the graph G for k > 1.
Theorem 3. Let G be a graph isomorphic to the line graph of the subdivision graph of C n {P k+1 }. Then Proof. The edge partitions, according to the degree sum of neighbor vertices of every vertex for the cases k = 1, k = 2, and k > 2, are shown in Tables 3-5, respectively. Using Equations (5) and (7), we obtain the required results. (S u , S v ) where uv ∈ E(G) (3,7) (7,9) (9,9) Number of edges n 2n 2n Number of edges n n n 2n 2n Let S m+1 be star graph on m + 2 vertices. The cluster product of C n by S m+1 , denoted by C n {S m+1 }, is obtained by identifying any pendent vertex of the i-th copy of S m+1 to the i-th vertex of C n . It is easy to see that there are mn + 2n vertices and edges in C n {S m+1 }. The graph of C n {S m+1 } is shown in Figure 2.  Then M α (G) = mn + n · 3 α+1 + n(m + 1) α+1 .
Proof. The structure of G is manifested in Figure 2. There is a total 2mn + 4n vertices among which the number of vertices with degree 1, 3, and m + 1 are mn, 3n, and mn + n, respectively. Thus, the expression of M α (G) is deduced by means of (2).

Theorem 5.
Let G be a graph isomorphic to the line graph of the subdivision graph of C n {S m+1 }. Then

ABC(G)
Proof. It is noted that there are 1 2 (m 2 n + 3mn + 10n) edges in G. Its edge partition, follow from the degrees of the vertices are depicted in Table 6. From (1), (3), (4), and (6), we get the required results.

i-th Vertex Rooted Product of Graphs
Motivated by the definition of rooted product of graphs, we define the i-th vertex rooted product of graphs. Let H be a labeled graph with order n and G be a sequence of k rooted graphs G 1 , G 2 , . . . , G k . Then the i-th vertex rooted product of H by G, denoted by H i {G 1 , G 2 , . . . , G k } is generated by identifying the root vertex of every G l to the i-th vertex of H for all l ∈ {1, 2, . . . , k}. In the special case when the components G 1 , G 2 , . . . , G k are mutually isomorphic to a graph L, the i-th vertex rooted product of H by G is denoted by H i,k {L} and so-called the i-th vertex cluster product of H by L.
Topological Indices of Line Graph of the Subdivision Graph of C i,r {P k+1 } Let C = C n be cycle on n vertices and let C i,r {P k+1 } be the i-th vertex cluster product of C n by P k+1 , where P k+1 is a path on k + 1 vertices. Obviously, there are n + kr vertices and edges in C i,r {P k+1 }. The graph of C i,r {P k+1 } is shown in Figure 3. Theorem 7. Let G be a graph isomorphic to the line graph of the subdivision graph of C i,r {P k+1 }. Then M α (G) = r + (n + kr − r − 2)2 α+1 + (r + 2) α+1 .
Proof. The basic structure of G is depicted in Figure 3. There are in total 2n + 2kr vertices, among which r vertices meet degree 1, 2n + 2kr − 2r − 2 vertices have degree 2, and the remaining r + 2 vertices satisfy degree r + 2. Therefore, the fomula for M α (G) is got in terms of (2). Theorem 8. Let G be a graph isomorphic to the line graph of the subdivision graph of C i,r {P k+1 }. Then

2.
Proof. By simple calculation the total number of edges of G are 1 2 (4n + r 2 + 4kr + r). The edge partitions, using the degrees of the vertices for the cases k = 1 and k > 1, are shown in Tables 8 and 9, respectively. Equations (1), (3), (4), and (6) implies the required results.

Conclusions
In this article, certain degree-based topological invariants, namely general sum-connectivity index, ABC 4 , generalized Randić index, GA 5 , and general Zagreb index for the line graphs of subdivision graphs of some classes of rooted product of graphs were studied for the first time. We have also constructed a new class of rooted product of graphs called i-th vertex rooted product of graph G by using the same concept of identifying the root vertex of sequence of graphs with the i-th vertex of G. Finally, in the last section we have computed the topological indices of the line graph of subdivision graph ith vertex cluster product of cycle of length r with path of length k.