On certain topological indices of the line graph of subdivision graphs

Applied Mathematics and Computation 271 (2015) 790–794

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

On certain topological indices of the line graph of subdivision graphs Muhammad Faisal Nadeem a, Sohail Zafar b,∗, Zohaib Zahid b a b

Department of Mathematics, COMSATS Institute of Information Technology, Lahore, Pakistan University of Management and Technology, Lahore, Pakistan

a r t i c l e

i n f o

Keywords: ABC4 index GA5 index Line graph Subdivision graph

a b s t r a c t In QSAR/QSPR study, topological indices such as Shultz index, generalized Randic index, Zagreb index, general sum-connectivity index, atom-bond connectivity (ABC) index and geometric-arithmetic (GA) index are utilized to guess the bioactivity of chemical compounds. A topological index in fact relates a chemical structure with a numeric number. Graph theory has established a significant use in this area of research. In this paper we computed ABC4 and GA5 indices of the line graph of tadpole, wheel and ladder graphs using the notion of subdivision. © 2015 Elsevier Inc. All rights reserved.

1. Introduction and preliminaries In this article we will consider only simple graphs without loop and multiple edges. Let G be a simple graph, with vertex  set V(G) and edge set E(G). The degree du of a vertex u is the number of edges that are incident to it and Su = v∈Nu dv where Nu = {v ∈ V (G)|uv ∈ E (G)}. Nu is also known as the set of neighbor vertices of u. For any natural number d, we define Vd = {u ∈ V (G) | Su = d}. Topological indices are the numerical quantities which represent the structure of any simple finite graph. They are invariant under the graph isomorphisms. The importance of topologically indices is generally linked with quantitative structure property relationship (QSPR) and quantitative structure activity relationship (QSAR) (see [18]). The first degree-based connectivity index for graphs developed by using vertex degrees is Randic index [15]. The Randic index of a graph G is defined as



R(G) =

(du dv )−1/2 .

uv∈E (G)

Later, this index was generalized for any real number α , and known as the generalized Randic index Rα (G):

Rα (G) =



(du dv )α .

uv∈E (G)

In 2010, general sum-connectivity index χ α (G) has been introduced in [26]:

χα (G) =



(du + dv )α .

uv∈E (G)

∗

Corresponding author. Tel.: +92 3214242349. E-mail addresses: [email protected] (M.F. Nadeem), [email protected] (S. Zafar), [email protected] (Z. Zahid).

http://dx.doi.org/10.1016/j.amc.2015.09.061 0096-3003/© 2015 Elsevier Inc. All rights reserved.

M.F. Nadeem et al. / Applied Mathematics and Computation 271 (2015) 790–794

791

The atom–bond connectivity (ABC) index, introduced by Estrada et al. in [3]. The ABC index of graph G is defined as





ABC (G) =

uv∈E (G)

du + dv − 2 . du dv

Vukicevic and Furtula introduced the geometric–arithmetic (GA) index in [24]. The GA index for graph G is defined by



 2 du dv GA(G) = . d u + dv uv∈E (G)

The fourth member of the class of ABC index was introduced by Ghorbani and Hosseinzadeh [6] as:

ABC4 (G) =





uv∈E (G)

Su + Sv − 2 Su Sv

(1)

The fifth GA index was introduced by Graovac et al. in [7] as

√  2 Su S v GA5 (G) = . Su + Sv

(2)

uv∈E (G)

For a collection of recent results on degree-based topological indices, we refer the interested reader to the articles [1,2,4,5, 8–14,19,21–23,25]. Now we define some notations of the graph theory. The subdivision graph S(G) is the graph obtained from G by replacing each of its edge by a path of length 2. The line graph L(G) of graph G is the graph whose vertices are the edges of G, two vertices e and f are incident if and only if they have a common end vertex in G. The tadpole graph Tn, k is the graph obtained by joining a cycle of n vertices with a path of length k. A ladder Ln is obtain by taking Cartesian product of two paths Pn × P2 . A wheel graph Wn of order n composed of a vertex, which will be called the hub, adjacent to all vertices of a cycle of order n i.e., Wn = Cn + K1 . The cycle will be called the rim of the wheel, and the edges connecting the hub to the vertices of the rim will be called the spokes. 2. Topological indices of L(S(G)) In 2011, Ranjini et al. calculated the explicit expressions for the Shultz index of the subdivision graphs of the tadpole, wheel, helm and ladder graphs [17]. They also studied the Zagreb indices of the line graph of the tadpole, wheel and ladder graphs with subdivision in [16]. In 2015, Su and Xu calculate the general sum-connectivity index and co-index of the L(S(Tn, k )), L(S(Wn )) and L(S(Ln )) in [20]. Motivated by the results of [20], we computed ABC4 and GA5 indices of the line graph of the tadpole, wheel and ladder graphs using the notion of subdivision. 2.1. ABC4 and GA5 indices of L(S(Tn, k )) Theorem 2.1. Let G be the line graph of subdivision graph of the tadpole graph. Then ABC4 index is given by

√ √ √ √ √ ⎧√ 2 42 14 182 110 (2n − 5) 6 35 ⎪ + + + + + , ⎪ ⎪ 8 21 14 10 4 5 ⎪ ⎪ √ √ √ √ √ ⎨√ 3 110 3 14 2 10 (2n − 5) 6 35 ABC4 (G) = + + + + + , ⎪ 2 5 20 8 4 5 ⎪ ⎪√ √ √ √ √ √ ⎪ ⎪ 3 110 3 14 15 (n + k − 5) 6 3 35 ⎩ 2 2

+

6

+

20

+

8

+

+

2

10

if k = 1; if k = 2; ,

if k > 2.

Proof. First consider the line graph L(S(Tn, k )) with k > 2. In this graph there are total 2(n + k) vertices, consisting 3 vertices of degree 3, one vertex of degree 1 and remaining all the vertices of degree 2. Therefore |V2 | = 1, |V3 | = 1, |V5 | = 3, |V8 | = 3 and |V4 | = 2(n + k − 4) as shown in Fig. 1(b). Hence, we get the edge partition, based on the degree sum of neighbor vertices of each vertex as shown in Table 3. We apply Formula (1) to Table 3 and obtain the required result. By similar arguments we can obtain the expressions of ABC4 (G) index for k = 1 and 2 by viewing edge partitions in Tables 1 and 2, respectively.  Table 1 The edge partition of L(S(Tn, k )) for k = 1.

(Su , Sv ) where uv ∈ E (G)

(3,7)

(5,8)

(7,8)

(8,8)

(4,4)

(4,5)

Number of edges

1

2

2

1

2n − 5

2

792

M.F. Nadeem et al. / Applied Mathematics and Computation 271 (2015) 790–794

(a)

(b)

5

2

4

4

4

8

4

8

4

8 3

4

4

4 5 5

4

Fig. 1. (a) The subdivision graph S(Tn, k ) of tadpole graph Tn, k ; (b) the line graph L(S(Tn, k )) labeled with degree sum of neighbor vertices. Table 2 The edge partition of L(S(Tn, k )) for k = 2.

(Su , Sv ) where uv ∈ E (G)

(2,3)

(3,5)

(5,8)

(8,8)

(4,4)

(4,5)

Number of edges

1

1

3

3

2n − 5

2

Table 3 The edge partition of L(S(Tn, k )) for k > 2.

(Su , Sv ) where uv ∈ E (G)

(2,3)

(3,4)

(4,4)

(4,5)

(5,8)

(8,8)

Number of edges

1

1

2(n + k − 5)

3

3

3

Table 4 The edge partition of L(S(Wn )).

(Su , Sv ) where uv ∈ E (G)

(9,9)

(9, n + 6)

(n + 6, n2 − n + 3)

(n2 − n + 3, n2 − n + 3)

Number of edges

2n

2n

n

n(n − 1)/2

Theorem 2.2. Let G be the line graph of subdivision graph of the tadpole graph Tn, k . Then GA5 index is

√ √ ⎧√ 8 10 8 14 21 ⎪ + + − 4 + 2n + ⎪ ⎪ 5 13 15 ⎪ ⎪ √ √ ⎨ √

√ 8 5 , 9 √ GA5 (G) = 2 6 + 15 + 12 10 − 2 + 2n + 8 5 , ⎪ 5 4 13 9 ⎪ ⎪ √ √ √ √ ⎪ ⎪ 4 5 ⎩ 2 6 4 3 12 10 + + − 7 + 2n + 2k + , 5 7 13 3

if k = 1; if k = 2; if k > 2.

Proof. Apply Formula (2) to the edge partitions shown in Tables 1– 3 to get the required results.  2.2. ABC4 and GA5 indices of L(S(Wn )) Theorem 2.3. Let G be the line graph of subdivision graph of the wheel graph Wn , then

ABC4 (G) =

2n 8n + 9 3



13 + n +n n+6



n(n − 1)  2 n2 + 7 2n − 2n + 4. + 2 (n + 6)(n − n + 3) 2(n2 − n + 3)

Proof. The graph L(S(Wn )) contains 4n vertices among which 3n vertices are of degree 3 and remaining all are of degree n. Therefore, |V9 | = 2n, |Vn+6 | = n and |Vn2 −n+3 | = n as shown in Fig. 2(b). Hence, we get the edge partition, based on the degree sum of neighbor vertices of each vertex as shown in Table 4. We apply Formula (1) to Table 4 and obtain the required result.  Theorem 2.4. Let G be the line graph of the subdivision graph of the wheel graph Wn . Then

 √ 2n (n + 6)(n2 − n + 3) 12n n + 6 n(n − 1) + . GA5 (G) = 2n + + n + 15 2 n2 + 9

Proof. Apply Formula (2) to the edge partition shown in Table 4 to get the required result.



M.F. Nadeem et al. / Applied Mathematics and Computation 271 (2015) 790–794

(a)

(b)

n-n+3

9

9

9

2

793

9

n+6

n+6

9

9 9

n+6

9

n+6 9

n+6 9 9

n+6 n+6

9

9

9

Fig. 2. (a) The subdivision graph S(Wn ) of the wheel graph Wn ; (b) the line graph L(S(Wn )) labeled with degree sum of neighbor vertices.

(a)

(b)

5

8

9

9

9

9

9

9

8

5

4

9

9

9

9

4

4

9

9

9

9

4

5

8

9

9

9

9

9

9

8

5

Fig. 3. (a) The subdivision graph S(Ln ) of ladder graph Ln ; (b) the line graph L(S(Ln )) labeled with degree sum of neighbor vertices.

Table 5 The edge partition of L(S(Ln )) for n = 3.

(Su , Sv ) where uv ∈ E (G)

(4,4)

(4,5)

(5,8)

(8,8)

(8,9)

(9,9)

Number of edges

2

4

4

2

4

1

Table 6 The edge partition of L(S(Ln )) for n > 3.

(Su , Sv ) where uv ∈ E (G)

(4,4)

(4,5)

(5,8)

(8,9)

(9,9)

Number of edges

2

4

4

8

9n−28

2.3. ABC4 and GA5 indices of L(S(Ln ))

Theorem 2.5. Let G be the line graph of subdivision graph of the ladder graph Ln . Then

ABC4 (G) =

√ √ √ √ ⎧√ 35 6 110 14 30 4 ⎪ ⎨ +2 + + + + ,

2 5 5 4 3 9 √ √ √ √ ⎪ ⎩ 6 + 2 35 + 110 + 2 30 + 4n − 112 , 2 5 5 3 9

if n = 3; if n > 3.

Proof. The ladder graph L2 and L(S(L2 )) are cycles of length 4 and 8 respectively, which is known in [6]. Consider L(S(Ln )) for the case n > 3. In this graph we have 6n − 4 vertices out of which 8 vertices are of degree 2 and all the remaining vertices are of degree 3. Therefore, |V4 | = 4, |V5 | = 4, |V8 | = 4 and |V9 | = 2(3n − 8) as shown in Fig. 3(b). Hence we get the edge partition based on the degree sum of neighbor vertices of each vertex as shown in Table 6. We apply Formula (1) to Table 6 and obtain the required result. Similar arguments will work for the case n = 3 (see Table 5). 

794

M.F. Nadeem et al. / Applied Mathematics and Computation 271 (2015) 790–794

Theorem 2.6. Let G be the line graph of subdivision graph of the ladder graph Ln . Then

GA5 (G) =

√ √ √ ⎧ 16 5 16 10 48 2 ⎪ ⎨5 + + + , 9

13 17 √ √ √ ⎪ 16 96 2 5 10 16 ⎩−26 + + + + 9n, 9 13 17

if n = 3; if n > 3.

Proof. Apply Formula (2) to the edge partitions shown in Tables 5 and 6 to get the required result.



3. Conclusion In this paper certain degree based topological indices, namely atomic-bond connectivity (ABC) index and geometricarithmetic (GA) index were studied for the case of line graph of the subdivision graphs. In future, we will pay attention to some new classes of line graph of subdivision graph and study their topological indices which will be reasonably useful to recognize their underlying topologies. Acknowledgment The authors are grateful to the reviewers for suggestions to improve the presentation of the manuscript. References [1] H. Abdo, D. Dimitrov, T. Reti, D. Stevanovic, Estimating the spectral radius of a graph by the second Zagreb index, MATCH Commun. Math. Comput. Chem. 72 (3) (2014) 741–751. [2] M.B. Ahmadi, D. Dimitrov, I. Gutman, S.A. Hosseini, Disproving a conjecture on trees with minimal atom–bond connectivity index, MATCH Commun. Math. Comput. Chem. 72 (3) (2014) 685–698. [3] E. Estrada, L. Torres, L. Rodriguez, I. Gutman, An atom–bond connectivity index: modelling the enthalpy of formation of alkanes, Indian J. Chem. 37A (1998) 849–855. [4] A. Hamzeh, T. Reti, An analogue of Zagreb index inequality obtained from graph irregularity measures, MATCH Commun. Math. Comput. Chem. 72 (3) (2014) 669–683. [5] S. Hosseini, M. Ahmadi, I. Gutman, Kragujevac trees with minimal atom–bond connectivity index, MATCH Commun. Math. Comput. Chem. 71 (2014) 5–20. [6] M. Ghorbani, M.A. Hosseinzadeh, Computing abc4 index of nanostar dendrimers, Optoelectron. Adv. Mater. Rapid Commun. 4 (9) (2010) 1419–1422. [7] A. Graovac, M. Ghorbani, M.A. Hosseinzadeh, Computing fifth geometric–arithmetic index for nanostar dendrimers, J. Math. Nanosci. 1 (2011) 33–42. [8] I. Gutman, An exceptional property of first Zagreb index, MATCH Commun. Math. Comput. Chem. 72 (2014) 733–740. [9] I. Gutman, B. Furtula, C. Elphick, Three new/old vertex-degree-based topological indices, MATCH Commun. Math. Comput. Chem. 72 (2014) 617–632. [10] R. Kazemi, The second Zagreb index of molecular graphs with tree structure, MATCH Commun. Math. Comput. Chem. 72 (3) (2014) 753–760. [11] X. Li, Y. Shi, A survey on the Randic´ index, MATCH Commun. Math. Comput. Chem. 59 (1) (2008) 127–156. [12] H. Lin, Vertices of degree two and the first Zagreb index of trees, MATCH Commun. Math. Comput. Chem. 72 (2014) 825–834. [13] J. Palacios, A resistive upper bound for the ABC index, MATCH Commun. Math. Comput. Chem. 72 (2014) 709–713. [14] J. Rada, R. Cruz, Vertex-degree-based topological indices over graphs, MATCH Commun. Math. Comput. Chem. 72 (2014) 603–616. [15] M. Randic, On characterization of molecular branching, J. Am. Chem. Soc. 97 (1975) 6609–6615. [16] P.S. Ranjini, V. Lokesha, I.N. Cangül, On the Zagreb indices of the line graphs of the subdivision graphs, Appl. Math. Comput. 218 (2011) 699–702. [17] P.S. Ranjini, V. Lokesha, M.A. Rajan, On the Shultz index of the subdivision graphs, Adv. Stud. Contemp. Math. 21 (3) (2011) 279–290. [18] M. Saheli, H. Saati, A.R. Ashrafi, The eccentric connectivity index of one pentagonal carbon nanocones, Optoelectron. Adv. Mater. Rapid Commun. 4 (6) (2010) 896–897. [19] Y. Shi, Note on two generalizations of the Randic index, Appl. Math. Comput. 265 (2015) 1019–1025. [20] G. Su, L. Xu, Topological indices of the line graph of subdivision graphs and their Schur-bounds, Appl. Math. Comput. 253 (2015) 395–401. [21] R.M. Tache, General sum-connectivity index with α ≥ 1 for bicyclic graphs, MATCH Commun. Math. Comput. Chem. 72 (3) (2014) 761–774. [22] I. Tomescu, M.K. Jamil, Maximum general sum-connectivity index for trees with given independence number, MATCH Commun. Math. Comput. Chem. 72 (3) (2014) 715–722. [23] A. Vasilyev, R. Darda, D. Stevanovic, Trees of given order and independence number with minimal first Zagreb index, MATCH Commun. Math. Comput. Chem. 72 (2014) 775–782. [24] D. Vukicevic, B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46 (2009) 1369–1376. [25] K. Xu, K.C. Das, S. Balachandran, Maximizing the Zagreb indices of (n, m)-graphs, MATCH Commun. Math. Comput. Chem. 72 (2014) 641–654. [26] B. Zhou, N. Trinajstic, On general sum-connectivity index, J. Math. Chem. 47 (2010) 210–218.