#### Abstract

A topological index is a numerical measure that characterises the whole structure of a graph. Based on vertex degrees, the idea of an atom-bond connectivity index was introduced in chemical graph theory. Later, different versions of the ABC index were created, and some of these indices were recently designed. In this paper, we present the edge version of the atom-bond connectivity index, edge version of the multiplicative atom-bond connectivity index, and atom-bond connectivity temperature () index for the line graph of subdivision graph of tadpole graph , ladder graph , and wheel graph . Numerical simulation has also been shown for some novel families of atom-bond connectivity index comparing the three types of indices which can be useful for QSAR and QSPR studies.

#### 1. Introduction

In this article, we have considered simple graphs, which are unweighted, undirected graphs that have no loops and multiple edges attached. Let be a simple graph, with vertex set and edge set . Suppose is an edge of , which connects the vertices and , then we denote and state that “ and are adjacent.” The degree of a vertex is the number of edges that are incident to it. Topological indices are the mathematical measures that correspond to the structure of any simple finite graph. They are invariant under the graph isomorphism. There are some famous degree-based topological indices, which are introduced and applied in chemical engineering, for instance, the Randic index (refer to Ali & Du , Li & Shi , and Shi  for more details). These indices are also significant in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) (see [4, 5]).

The subdivision graph [6, 7] is the graph obtained from by replacing each of its edge by a path of length 2. In graph , if the corresponding edges share a vertex in , the line graph of a graph is considered as a graph with vertices of the edges in . Two vertices and are incident if and only if they have a common end vertex in . Estrada et al.  put forward a topological index named atom-bond connectivity index (briefly, ) aswhere and represent the degrees of the vertices and , respectively. Recent advances on index can be referred to in Das et al. , Lin et al. , Gao & Shao , and Bianchi et al. . Referring to the end vertex degree and of edges and in a line graph of , Farahani  proposed the edge version of atom-bond connectivity, index. This idea is described in the following:where and are the degree of the edge and , respectively. The reader can find more information about index in References .

The multiplicative atom-bond connectivity index was introduced by Kulli in 2016 . More information regarding the multiplicative atom-bond connectivity index may be found in references [20, 21]. Later, the edge version of the multiplicative atom-bond connectivity index  of a graph was introduced, and it is defined aswhere is the degree of the edge in .

The temperature of a vertex of a connected graph is defined by Fajtlowicz  aswhere is the degree of a vertex and is the size of a graph . Recently, Kahasy et al.  introduced a new index, known as the atom-bond connectivity temperature index. This index is defined as follows:where and are the temperature of the vertex and , respectively.

#### 2. Main Results

In 2011, Ranjini et al. calculated the explicit expression for the Shultz indices of the subdivision graphs of the tadpole, wheel, helm, and ladder graphs . They also studied the Zagreb indices of line graphs of tadpole, wheel, and ladder graphs with subdivision in . In 2015, Su & Xu calculated the general sum-connectivity indices and coindices of line graphs of tadpole, wheel, and ladder graphs with subdivision in . In , Nadeem et al. computed and indices of the line graphs of these graphs by using the notion of subdivision. They also studied the and of these graphs . Other studies on these include Rajasekar & Nagarajan  research on the location domination number of the line graph. Recently, Li &. Taylor  also studied the first Zagreb index and some Hamilton properties of the line graph.

A tadpole graph is the graph obtained by joining a cycle of vertices with a path of length . A ladder is obtained by taking the Cartesian product of two paths . A wheel graph or order composed of a vertex is called the hub, adjacent to all vertices of a cycle of the order .

Motivated by the results of [26, 27, 31], we studied the line graph of the subdivision graph , , and and derived an expression for the edge version of atom-bond connectivity, multiplicative atom-bond connectivity indices, and atom-bond connectivity temperature index of the graphs , , and .

Theorem 1. The edge version of the atom-bond connectivity index of is

Proof. Let be the line graph of the subdivision graph , seeing Figure 1. It contained edges of the subdivision graph of , and then, in the graph of , it contained vertices. It consists of three types of degree of edge , such as , , and . Out of which, vertices are of the degree , one vertex of degree and the remaining vertices are of the degree . The graph of contains path of length . Let be the vertex of degree which is attached to this path. Let and be the neighbor of which are of degree in the . The vertices and have two neighbors of degree and one neighbor of degree in , where is the edge adjacent to . The vertex has adjacent vertices of degree and one vertex of degree in the path. Let we derive an expression for the edge version of topological indices of the graph , for . In graph , it contains a path of length which attached with . Hence, with respect to the path is . For corresponding to the vertices , in , hence, we have . Since one edge in is shared between pairs of vertices, . Among the remaining, vertices, for vertices, have neighbors of degree , and one vertex has neighbor of degree Hence, with respect to vertices is . Adding all these number together, the edge version of atom-bond connectivity index of is found as .
If , then the graph contains path of length which attached with . Hence, with respect to the path is . For corresponding to the vertices , in , hence, we have . Since one edge in is shared between pairs of vertices, . Out of vertices, for vertices have neighbors of degree and one vertex has neighbor of degree Hence, with respect to vertices is . Adding all these number together, the edge version of atom-bond connectivity index of is found as . This completes the proof.

Theorem 2. The edge version of the multiplicative atom-bond connectivity index of is

Proof. After adopting the induction method, it is clear that overall speaking, this line graph of subdivision graph possesses of vertices and edges. If and are the degree of edge , then there are edge of type , , edges with , edges of type , , edges with . Hence, for graph with , we have . For , we have edge of type , , edges with , edges of type , , edges with . Hence, we deduceThis completes the proof.

Theorem 3. The atom-bond connectivity temperature index of is

Proof. In , there are total vertices, among which vertices are of the degree , one vertex of degree and the remaining vertices are of the degree . The total number of edges of is . For , we have vertices, which one vertex of degree , vertices of degree and vertices of degree . Therefore, after adopting the induction trick, we have edge partition based on the temperature. If and are the temperature of the vertex and , then there are edge of type , , edges with , edges of type , , edges with . Hence, for graph with , we haveFor , we have edge of type , , edges with , edges of type , , edges with . Therefore, we getThis completes the proof.

Theorem 4. The edge version of the atom-bond connectivity index of is

Proof. Let be the line graph of the subdivision graph , seeing Figure 2. It contains vertices are of degree and vertices of degree . Out of edges, the edges of degree have neighbor of degree . Hence, corresponding to these edges which have only neighbor of degree is . The remaining vertices of degree are adjacent to vertices of degree . Hence, with respect to these vertices is . Also remaining n(n-1)/2 edges of degree n have neighbor of degree n. Hence, with respect to all these degrees of is . Adding all these number together, the edge version of atom-bond connectivity index of is found as . This completes the proof.

Theorem 5. The edge version of the multiplicative atom-bond connectivity index of is

Proof. After adopting the induction technology, it is clear to find that, roughly speaking, this line graph of subdivision graph has contained vertices and edges. Also, there are edges of type , edges of type , , edges with . As a result, we inferThis completes the proof.

Theorem 6. The atom-bond connectivity temperature index of is

Proof. In , there are total vertices are of degree and vertices of degree . The total number of edges of is . After adopting the induction method, we have edge partition based on the temperature. If and are the temperature of the vertex and , then there are edge of type , edges with , and edges of type . Hence, we deduceThis completes the proof.

Theorem 7. The edge version of the atom-bond connectivity index of is

Proof. Let be the line graph of subdivision graph , seeing Figure 3. The number of vertices in is among which vertices are of degree and the remaining vertices are of degree . The number of edges in is among which edges are of degree with itself, edges are of degree and , and the remaining edges are of degree 3 with itself. Adding all these numbers together, the edge version of the atom-bond connectivity index of is found as . This completes the proof.

Theorem 8. The edge version of multiplicative atom-bond connectivity index of is

Proof. After adopting the induction trick, we can find that, in general, this line graph of subdivision graph has vertices and edges. At the same, there are edges of type , edges of type , , edges with . Therefore, we getThis completes the proof.

Theorem 9. The atom-bond connectivity temperature index of is

Proof. In , there are total vertices in which vertices are of degree and the remaining vertices are of degree . The total number of edges of is . After adopting the induction technology, we have edge partition based on the temperature. If and are the temperature of the vertex and , then there are edge of type , edges with , and edges of type . As a result, we inferThis completes the proof.

#### 3. Numerical Simulation and Conclusion

In this paper, we propose some novel families of atom-bond connectivity index. Now, the results of these indices will be compared. The comparison between , , and of is shown in Table 1. The graphical representation of Table 1 is illustrated in Figure 4.

Similarly, the results for , , and of are compared in Table 2. Table 2 is given in Figure 5. Finally, the comparison of , , and of is shown in Table 3. The illustration of the results for , , and is shown as Figure 6.

In this paper, certain degree-based topological indices, namely, indices, were studied for the case of the line graphs of the subdivision graphs. It is anticipated that this computational study will encourage the researchers to have a firm grasp on the index framework they have chosen. The computational technique presented here can be useful for analysing the physicochemical features of the specified network, as well as being cost-effective and time-efficient. Future work includes the investigation of new classes of line graph of subdivision graphs and their topological indices which is useful in QSAR and QSPR studies.

#### Data Availability

All data required for this paper are included within these papers.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The authors acknowledge Universiti Malaysia Terengganu for providing the financial support under Talent and Publication Enhancement-Research Grant (TAPE-RG; vot no. 55330).