# Investigation of Atom-Bond Connectivity Indices of Line Graphs Using Subdivision Approach

Authors: Mohamad Nazri Husin; Sohail Zafar; R. U. Gobithaasan
Journal: Mathematical Problems in Engineering
 (2022)

---

## 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 connectivityABC 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 ABCe index, edge version of the multiplicative atom-bond connectivity ABCIIe index, and atom-bond connectivity temperature (ABCT) index for the line graph of subdivision graph of tadpole graph Tn,k, ladder graph Ln, and wheel graph Wn+1. 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.


---

## Body

## 1. Introduction

In this article, we have considered simple graphs, which are unweighted, undirected graphs that have no loops and multiple edges attached. LetG be a simple graph, with vertex set VG and edge set EG. Suppose e is an edge of G, which connects the vertices u and v, then we denote e=uv and state that “u and v are adjacent.” The degree du of a vertex u 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 [1], Li & Shi [2], and Shi [3] 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] SG is the graph obtained from G by replacing each of its edge by a path of length 2. In graph G, if the corresponding edges share a vertex in G, the line graph LG of a graph G is considered as a graph with vertices of the edges in G. Two vertices e and f are incident if and only if they have a common end vertex in G. Estrada et al. [8] put forward a topological index named atom-bond connectivity index (briefly, ABC) as(1)ABCG=∑uv∈EGdu+dv−2du×dv,where du and dv represent the degrees of the vertices u and v, respectively. Recent advances on ABC index can be referred to in Das et al. [9], Lin et al. [10], Gao & Shao [11], and Bianchi et al. [12]. Referring to the end vertex degree de and df of edges e and f in a line graph of G, Farahani [13] proposed the edge version of atom-bond connectivity, ABCe index. This idea is described in the following:(2)ABCeG=∑ef∈EGde+df−2de×df,where de and df are the degree of the edge e and f, respectively. The reader can find more information about ABCe index in References [14–18].The multiplicative atom-bond connectivity index was introduced by Kulli in 2016 [19]. 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 [22] of a graph G was introduced, and it is defined as(3)ABCIIeG=∏ef∈ELGde+df−2de⋅df,where de is the degree of the edge e in LG.The temperature of a vertexu of a connected graph G is defined by Fajtlowicz [23] as(4)Tu=dun−du,where du is the degree of a vertex u and n is the size of a graph G. Recently, Kahasy et al. [24] introduced a new index, known as the atom-bond connectivity temperature index. This index is defined as follows:(5)ABCTG=∏uv∈EGTu+Tv−2TuTv,where Tu and Tv are the temperature of the vertex u and v, 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 [25]. They also studied the Zagreb indices of line graphs of tadpole, wheel, and ladder graphs with subdivision in [26]. In 2015, Su & Xu calculated the general sum-connectivity indices and coindices of line graphs of tadpole, wheel, and ladder graphs with subdivision in [27]. In [28], Nadeem et al. computed ABC4 and GA5 indices of the line graphs of these graphs by using the notion of subdivision. They also studied the ABC4 and GA5 of these graphs [28]. Other studies on these include Rajasekar & Nagarajan [29] research on the location domination number of the line graph. Recently, Li &. Taylor [30] also studied the first Zagreb index and some Hamilton properties of the line graph.A tadpole graphTn,k is the graph obtained by joining a cycle of n vertices with a path of length k. A ladder Ln is obtained by taking the Cartesian product of two paths Pn×P2. A wheel graph Wn or order n composed of a vertex is called the hub, adjacent to all vertices of a cycle of the order n.Motivated by the results of [26, 27, 31], we studied the line graph of the subdivision graph Tn,k, Ln, and Wn+1 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 LSTn,k, LSWn+1, and LSLn.Theorem 1.
The edge version of the atom-bond connectivity index ofLSTn,k is(6)ABCeG=2n+2k+2−2,whenk>1;2n+2+23−322,whenk=1.Proof.
LetG be the line graph of the subdivision graph LSTn,k, seeing Figure 1. It contained 2n+k edges of the subdivision graph of STn,k, and then, in the graph of G, it contained 2n+k vertices. It consists of three types of degree of edge e, such as 1, 2, and 3. Out of which, 3 vertices are of the degree 3, one vertex of degree 1 and the remaining 2n+k−2 vertices are of the degree 2. The graph of G contains path of length 2k−1. Let V1 be the vertex of degree 3 which is attached to this path. Let V1′ and V2′ be the neighbor of V1 which are of degree 3 in the LG. The vertices V1′ and V2′ have two neighbors of degree 3 and one neighbor of degree 2 in LSCn+e, where e is the edge adjacent to SCn. The vertex V1 has 2 adjacent vertices of degree 3 and one vertex of degree 2 in the path. Let we derive an expression for the edge version of topological indices of the graph G, for k=1. In graph G, it contains a path of length 1 which attached with V1. Hence, ∑de+df−2/de⋅df with respect to the path is 2/3. For ∑de+df−2/de⋅df corresponding to the vertices V1, V1′,andV2′ in LSCn+e, hence, we have 4+2. Since one edge in G is shared between pairs of vertices, ∑de+df−2/de⋅df=2+1/2. Among the remaining, 2n−4 vertices, for 2n−5 vertices, have neighbors of degree 2, and one vertex has neighbor of degree 3. Hence, ∑de+df−2/de⋅df with respect to 2n−4 vertices is 2n−22. Adding all these number together, the edge version of atom-bond connectivity index of G is found as ABCeG=2n+2+2/3−32/2.
Ifk>1, then the graph G contains path of length 2k−1 which attached with V1. Hence, ∑de+df−2/de⋅df with respect to the path is 1/22k−1. For ∑de+df−2/de⋅df corresponding to the vertices V1, V1′andV2′ in LSCn+e, hence, we have 4+2. Since one edge in G is shared between pairs of vertices, ∑de+df−2/de⋅df=2+1/2. Out of 2n−2 vertices, for 2n−3 vertices have neighbors of degree 2 and one vertex has neighbor of degree 3. Hence, ∑de+df−2/de⋅df with respect to 2n−2 vertices is 2n−2. Adding all these number together, the edge version of atom-bond connectivity index of G is found as ABCeG=2n+2k+2−2. This completes the proof.Figure 1
[27] (a) The subdivision graph STn,k of the tadpole graph Tn,k. (b) The line graph LSTn,k.
(a)(b)Theorem 2.
The edge version of the multiplicative atom-bond connectivity index ofLSTn,k is(7)ABCIIeG=32n+k−3,whenk>1;2232n−5,whenk=1.Proof.
After adopting the induction method, it is clear that overall speaking, this line graph of subdivision graph possesses of2n+k vertices and 2n+k+1 edges. If de and df are the degree of edge e, then there are 1 edge of type de=1, df=3, 2n−5 edges with de=df=2, 2 edges of type de=2, df=3, 3 edges with de=df=3. Hence, for graph G with k=1, we have ABCIIeG=22/32n−5. For k>1, we have 1 edge of type de=1, df=2, 2n+2k−6 edges with de=df=2, 3 edges of type de=2, df=3, 3 edges with de=df=3. Hence, we deduce(8)ABCIIeG=11+2−21×2×2n+2k−62+2−22×2×32+3−22×3×33+3−23×3=32n+k−3.
This completes the proof.Theorem 3.
The atom-bond connectivity temperature index ofLSTn,k is(9)ABCTG=−4n2+9n−8nk+9k−4k2−4+2n+2k−6−2n−2k+4n+k−1,+3−4n2−8nk+15n−4k2+15k−123+3−4n−4k+122n+2k−39,whenk>1;2n−5−2n2+2n+−8n2+8n+43+2−4n2+7n−13,+2−2n2+5n−2,whenk=1.Proof.
InG, there are total 2n+k vertices, among which 3 vertices are of the degree 3, one vertex of degree 1 and the remaining 2n+k−2 vertices are of the degree 2. The total number of edges of G is 2n+k+1. For k=1, we have 2n+1 vertices, which one vertex of degree 1, 3 vertices of degree 3 and 2n−1 vertices of degree 2. Therefore, after adopting the induction trick, we have 4 edge partition based on the temperature. If Tu and Tv are the temperature of the vertex u and v, then there are 1 edge of type T1=1/2n+1, T3=3/2n−1, 2n−5 edges with T2=T2=1/n, 2 edges of type T2=1/n, T3=3/2n−1, 3 edges with T3=T3=3/2n−1. Hence, for graph G with k=1, we have(10)ABCTG=11/2n+1+3/2n−1−21/2n+1×3/2n−1+2n−51/n+1/n−21/n×1/n+21/n+3/2n−1−21/n×3/2n−1+33/2n−1+3/2n−1−23/2n−1×3/2n−1=2n−5−2n2+2n+−8n2+8n+43+2−4n2+7n−13+2−2n2+5n−2.
Fork>1, we have 1 edge of type T1=1/2n+2k−1, T2=2/2n+2k−2, 2n+2k−6 edges with T2=T2=2/2n+2k−2, 3 edges of type T2=2/2n+2k−2, T3=3/2n+2k−3, 3 edges with T3=T3=3/2n+2k−3. Therefore, we get(11)BCTG=11/2n+2k−1+2/2n+2k−2−21/2n+2k−1×2/2n+2k−2+2n+2k−62/2n+2k−2+2/2n+2k−2−21/2n+2k−2×3/2n+2k−2+32/2n+2k−2+3/2n+2k−3−22/2n+2k−2×3/2n+2k−3+33/2n+2k−3+3/2n+2k−3−23/2n+2k−3×3/2n+2k−3=−4n2+9n−8nk+9k−4k2−4+2n+2k−6−2n−2k+4n+k−1+3−4n2−8nk+15n−4k2+15k−123+3−4n−4k+122n+2k−39.
This completes the proof.Theorem 4.
The edge version of the atom-bond connectivity index ofLSWn+1 is(12)ABCeG=8n3+nn+13n+n−12n−12.Proof.
LetG be the line graph of the subdivision graph LSTn,k, seeing Figure 2. It contains 4n vertices are of degree 3 and n vertices of degree n. Out of n2+9n/2 edges, the 4n edges of degree 3 have neighbor of degree 3. Hence, ∑de+df−2/de⋅df corresponding to these 4n edges which have only neighbor of degree 3 is 8n/3. The remaining n vertices of degree 3 are adjacent to vertices of degree n. Hence, ∑de+df−2/de⋅df with respect to these n vertices is nn+1/3n. Also remaining n(n-1)/2 edges of degree n have neighbor of degree n. Hence, ∑de+df−2/de⋅df with respect to all these degrees of n is n−12n−1/2. Adding all these number together, the edge version of atom-bond connectivity index of G is found as ABCeG=8n/3+nn+1/3n+n−12n−1/2. This completes the proof.Figure 2
[27] (a) The subdivision graph SWn of the tadpole graph Wn. (b) The line graph LSWn.
(a)(b)Theorem 5.
The edge version of the multiplicative atom-bond connectivity index ofLSWn+1 is(13)ABCIIeG=4n2n−13+n−2/3n2n−23.Proof.
After adopting the induction technology, it is clear to find that, roughly speaking, this line graph of subdivision graph has contained4n vertices and n2+9n/2 edges. Also, there are 4n edges of type de=df=3, n edges of type de=3, df=n, nn−1/2 edges with de=df=n. As a result, we infer(14)ABCIIeG=4n3+3−23×3×n3+n−23×n×nn−12n+n−2n×n=4n2n−13+n−2/3n2n−23.
This completes the proof.Theorem 6.
The atom-bond connectivity temperature index ofLSWn+1 is(15)ABCTG=43n4n−3−8n+12+n−20n+243+3nn−1.Proof.
InG, there are total 4n vertices are of degree 3 and n vertices of degree n. The total number of edges of G is n2+9n/2. After adopting the induction method, we have 3 edge partition based on the temperature. If Tu and Tv are the temperature of the vertex u and v, then there are 4n edge of type T3=T3=3/4n−3, n edges with T3=3/4n−3, Tn=1/3 and nn−1/2 edges of type Tn=Tn=1/3. Hence, we deduce(16)ABCTG=4n3/4n−3+3/4n−3−23/4n−3×3/4n−3+n3/4n−3+1/3−23/4n−3×1/3+nn−121/3+1/3−21/3×1/3=43n4n−3−8n+12+n−20n+243+3nn−1.
This completes the proof.Theorem 7.
The edge version of the atom-bond connectivity index ofLSLn is(17)ABCeG=52+29n−203.Proof.
LetG be the line graph of subdivision graph LSLn, seeing Figure 3. The number of vertices in G is 6n−4 among which 8 vertices are of degree 2 and the remaining 6n−12 vertices are of degree 3. The number of edges in G is 9n−10 among which 6 edges are of degree 2 with itself, 4 edges are of degree 2 and 3, and the remaining 9n−20 edges are of degree 3 with itself. Adding all these numbers together, the edge version of the atom-bond connectivity index of G is found as ABCeG=52+29n−20/3. This completes the proof.Figure 3
[27]: (a) The subdivision graph SLn of the tadpole graph Ln. (b) The line graph LSLn.
(a)(b)Theorem 8.
The edge version of multiplicative atom-bond connectivity index ofLSLn is(18)ABCIIeG=89n−20.Proof.
After adopting the induction trick, we can find that, in general, this line graph of subdivision graph has6n−4 vertices and 9n−10 edges. At the same, there are 6 edges of type de=df=2, 4 edges of type de=2, df=3, 9n−20 edges with de=df=3. Therefore, we get(19)ABCIIeG=62+2−22×2×42+3−22×3×9n−203+3−23×3=89n−20.
This completes the proof.Theorem 9.
The atom-bond connectivity temperature index ofLSLn is(20)ABCTG=63n−3−6n+8+4−36n2+93n−583+9n−206n−7−12n+203.Proof.
InG, there are total 6n−4 vertices in which 8 vertices are of degree 2 and the remaining 6n−12 vertices are of degree 3. The total number of edges of G is 9n−10. After adopting the induction technology, we have 3 edge partition based on the temperature. If Tu and Tv are the temperature of the vertex u and v, then there are 6 edge of type T2=T2=1/3n−3, 4 edges with T2=1/3n−3, T3=3/6n−7 and 9n−20 edges of type T3=T3=3/6n−7. As a result, we infer(21)ABCTG=61/3n−3+1/3n−3−21/3n−3×1/3n−3+41/3n−3+3/6n−7−21/3n−3×3/6n−7+9n−203/6n−7+3/6n−7−23/6n−7×3/6n−7=63n−3−6n+8+4−36n2+93n−583+9n−206n−7−12n+203.
This 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 betweenABCe, ABCIIe, and ABCT of LSTn,k is shown in Table 1. The graphical representation of Table 1 is illustrated in Figure 4.Table 1
Comparison betweenABCe, ABCe, and ABCT of LSTn,k.
n,kABCeABCIIeABCT(1, 1)2.1093898−4.8989794864.7876937(2, 2)6.2426406874.24264068723.98539667(3, 3)9.07106781212.7279220672.10624376(4, 4)11.8994949421.21320344142.5586461(5, 5)14.7279220629.69848481235.5731609(6, 6)17.5563491938.18376618351.1898721(7, 7)20.3847763146.66904756489.4215973(8, 8)23.2132034455.15432893650.2737199(9, 9)26.0416305663.63961031833.7488996(10, 10)28.8700576972.124891681039.848602Figure 4
Comparison betweenABCe, ABCe, and AABCe of LSTn,k.Similarly, the results forABCe, ABCIIe, and ABCT of LSWn+1 are compared in Table 2. Table 2 is given in Figure 5. Finally, the comparison of ABCe, ABCIIe, and ABCT of LSLn is shown in Table 3. The illustration of the results for ABCe, ABCIIe, and ABCT is shown as Figure 6.Table 2
Comparison betweenABCe, ABCIIe, and ABCT of LSWn+1.
nABCeABCIIeABCT13.48316324803.82136720527.4546536775.33333333320.008599653123262.35382907416.92289018101.1928851124.0640009522.15246524238.5139176205.2566258627.64735154473.2863826305.8662831733.37946531838.1312546425.8255499839.328113671368.662608565.0772788945.477225582103.254621723.57387221051.813850473082.855819901.2753388Figure 5
Comparison betweenABCe, ABCIIe, and ABCT of LSWn+1.Table 3
Comparison betweenABCe, ABCIIe, and ABCT of LSLn.
nABCeABCIIeABCT1−0.2622655215−88−8.06149838125.737734479−1627.04079003311.7377344856100.868826417.73773448128225.7405179523.73773448200401.5280011629.73773448272628.2266894735.73773448344905.8363724841.737734484161234.357204947.737734484881613.7893291053.737734485602044.132853Figure 6
Comparison betweenABCe, ABCIIe, and ABCT of LSLn.In this paper, certain degree-based topological indices, namely,ABC 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.



---

*Source: 6219155-2022-06-17.xml*