Some Computational Aspects for the Line Graph of Banana Tree Graph

Home , Tree (graph theory)

Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 2601-2627 © Research India Publications http://www.ripublication.com/gjpam.htm

Muhammad Saeed Ahmad Department of Mathematics, Government Muhammdan Anglo Oriental College, Lahore 54000, Pakistan.

Waqas Nazeer Division of Science and Technology, University of Education, Lahore 54000, Pakistan.

Shin Min Kang1 Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea.

Chahn Yong Jung Department of Business Administration, Gyeongsang National University, Jinju 52828, Korea.

Abstract A line graph has many useful applications in physical chemistry. M-polynomial is rich in producing closed forms of many degree-based topological indices which correlate chemical properties of the material under investigation. In this report, we compute closed form of the M-polynomial for the line graph of banana tree

1Corresponding author. graph. From the M-polynomial we recover some degree-based topological indices. Moreover, we plot our results.

AMS subject classiﬁcation: 05C12, 05C90. Keywords: M-polynomial, topological index, line graph.

1. Introduction

In chemical graph theory, a molecular graph is a simple graph (having no loops and multiple edges) in which atoms and chemical bonds between them are represented by vertices and edges respectively. A graph G(V, E), with vertex set V (G) and edge set E(G) is connected if there exists a connection between any pair of vertices in G. A network is simply a connected graph having no multiple edges and loops. The degree of a vertex is the number of vertices which are connected to that fixed vertex by the edges. The distance between two vertices u and v is denoted as d(u, v) = dG(u, v) and is defined as the length of shortest path between u and v in graph G. The number of vertices of G, adjacent to a given vertex v, is the “degree” of this vertex, and will be denoted by dv(G) or, if misunderstanding is not possible, simply by dv. The concept of degree is somewhat closely related to the concept of valence in chemistry. For details on bases of graph theory, any standard text such as [43] can be of great help. Cheminformatics is another emerging field in which quantitative structure-activity and Structure-property relationships predict the biological activities and properties of nano-material. In these studies, some Physico-chemical properties and topological indices are used to predict bioactivity of the chemical compounds see [5, 9, 26, 45, 39]. Algebraic polynomials have also useful applications in chemistry such as Hosoya polynomial (also called Wiener polynomial) [18] which play a vital role in determining distance- based topological indices. Among other algebraic polynomials, the M-polynomial [10] introduced in 2015, plays the same role in determining the closed form of many degree- based topological indices [1, 31, 32, 33, 34]. The main advantage of the M-polynomial is the wealth of information that it contains about degree-based graph invariants. The line graph L(G) of a graph G is the graph each of whose vertices, represents an edge of G and two of its vertices are adjacent if their corresponding edges are adjacent in G. In this article, we compute closed form of some degree-based topological indices of the line graph of Banana tree graph by using the M-polynomial. Some of these topological indices were calculated directly in [40].

2. Basic deﬁnitions and literature review

Here we give some basic deﬁnitions and literature review. Some Computational Aspects for the line graph of Banana Tree Graph 3

Deﬁnition 2.1. The M-polynomial of G is deﬁned as: i j M(G,x,y) = mij (G)x y , δ≤i≤j≤ where δ = min{dv : v ∈ V (G)},= max{dv : v ∈ V (G)}, and mij (G) the number of edges vu ∈ E(G) such that {dv,du}={i, j}.

Weiner [44] in 1947 approximated the boiling point of alkanes as αW(G)+βP3 +γ, where α, β and γ are empirical constants, W(G) is the Weiner index and P3 is the number of paths of length 3 in G. Thus Weiner laid the foundation of topological index which is also known as connectivity index. A lot of chemical experiments require determining the chemical properties of emerging nanotubes and nanomaterials. Chemical-based experiments reveal that out of more than 140 topological indices, no single index is strong enough to determine many physico-chemical properties, although, in combination, these topological indices can do this to some extent. The Wiener index is originally the first and most studied topological index, see for details [11, 21]. Randićindex, [36] denoted by R−1/2(G) and introduced by Milan Randićin 1975, is also one of the oldest topological indices. The Randićindex is defined as 1 R−1/2(G) = √ . d d uv∈E(G) u v In 1998, working independently, Bollobas and Erdos [4] andAmic et al. [2] proposed the generalized Randićindex and has been studied extensively by both chemists and mathematicians [23] and many mathematical properties of this index have been discussed in [6]. For a detailed survey we refer the book [27]. = 1 The general Randićindex is defined as: Rα(G) α , and the inverse (dudv) uv∈E(G) α Randićindex is defined as RRα(G) = (dudv) . Obviously, R−1/2(G) is the uv∈E(G) 1 particular case of R (G) when α =− . α 2 The Randićindex is a most popular, most often applied and most studied index among all topological indices. Many papers and books such as [24, 25, 27] are written on this topological index. Randićhimself wrote two reviews on his Randićindex [37, 38] and there are three more reviews on it, see [20, 28, 29]. The suitability of the Randićindex for drug design was immediately recognized, and eventually, the index was used for this purpose on countless occasions. The physical reason for the success of such a simple graph invariant is still an enigma, although several more-or-less plausible explanations were offered. Gutman and Trinajstic [22] introduced first Zagreb index and second Zagreb index, which are defined as: M1(G) = (du + dv) and M2(G) = (du × dv), uv∈E(G) uv∈E(G) respectively. For detail about these indices we refer [8, 19, 35, 41, 42] to the readers. 4 M. S. Ahmad, et al.

Both the first Zagreb index and the second Zagreb index give greater weights to the inner vertices and edges, and smaller weights to outer vertices and edges which oppose intuitive reasoning [30]. For a simple connected graph G, the second modified Zagreb index is defined as: 1 mM (G) = . 2 d(u)d(v) uvE(G)

The symmetric division index [SDD] is the one among 148 discrete Adriatic indices and is a good predictor of the total surface area for polychlorobiphenyls, see [17]. The symmetric division index of a connected graph G, is deﬁned as: min(d ,d ) max(d ,d ) SDD(G) = u v + u v . max(du,dv) min(du,dv) uv∈E(G)

Another variant of Randi´cindex is the harmonic index deﬁned as

2 H (G) = . du + dv uv∈E(G)

As far as we know, this index first appeared in [14]. Favaron et al. [15] considered the relation between the harmonic index and the eigenvalues of graphs. The inverse sum index, is the descriptor that was selected in [3] as a significant predictor of total surface area of octane isomers and for which the extremal graphs obtained with the help of mathematical chemistry have a particularly simple and elegant structure. The inverse sum index is defined as:

d d I (G) = u v . du + dv uv∈E(G)

The augmented Zagreb index of G proposed by Furtula et al. [16] is deﬁned as d d 3 A(G) = u v . du + dv − 2 uv∈E(G)

This graph invariant has proven to be a valuable predictive index in the study of heat of formation in octanes and heptanes (see [16]), whose prediction power is better than atom-bond connectivity index (please refer to [7, 13, 12] for its research background). Moreover, the tight upper and lower bounds for the augmented Zagreb index of chemical trees, and the trees with minimal augmented Zagreb index were obtained in [16]. The following Table 1 relates some well-known degree-based topological indices Some Computational Aspects for the line graph of Banana Tree Graph 5 with the M-polynomial [10].

Table 1. Derivation of topological indices topological indices derivation from M(G; x,y) first Zagreb index (Dx + Dy)(M(G; x,y))x=y=1 second Zagrab index (DxDy)(M(G; x,y))x=y=1 modified second Zagrab index (SxSy)(M(G; x,y))x=y=1 α α ; Randićindex (Dx Dy )(M(G x,y))x=y=1 α α ; inverse Randićindex (Sx Sy )(M(G x,y))x=y=1 symmetric division index (DxSy + SxDy)(M(G; x,y))x=y=1 harmonic index 2SxJ(M(G; x,y))x=y=1 Inverse Sum Index SxJDxDy(M(G; x,y))x=y=1 3 3 3 ; augmented Zagreb index Sx Q−2JDxDy(M(G x,y))x=y=1 In Table 1, ∂(f(x,y)) ∂(f(x,y)) Dx = x ,Dy = y , ∂x ∂y x f(t,y) y f(t,y) Sx = dt, Sy = dt, 0 t 0 t α J(f(x,y)) = f(x,x), Qα(f (x, y)) = x f(x,y). The following lemmas [19] will be helpful for our results. Lemma 2.2. Let G be a graph with u, v ∈ V (G) and e = uv ∈ E(G). Then

de = du + dv − 2.

Lemma 2.3. Let G be a graph of order p and size q. Then the line graph L(G) of G is 1 a graph of order p and size M (G) − q. 2 1

3. Results and discussions

In this part, we give or main computational results. The Banana tree graph Bn,k is the graph obtained by connecting one leaf of each of n copies of a k-star graph with a single root vertex that is distinct for all the stars. The Bn,k has order nk + 1 and size nk. B3,5 is shown in Figure 1.

Theorem 3.1. Let G be the line graph of Banana graph. Then the M-polynomial of G is n(n − 1) − − − M(G; x,y) = xnyn + nkk 1yn + ((k − 2)n)xk 1yk 2 2 nk2 + 6n − 5kn + xk−2yk−2. 2 6 M. S. Ahmad, et al.

Figure 1: The Banana tree graph B3,5

Figure 2: The line graph of Banana tree graph B3,5

Proof. The graph G for n = 3 and k = 5 is shown in Figure 2. By using Lemma 2.2, it is easy to see that the order of G is nk out of which (k − 2)n vertices are of degree k − 2,nvertices are of degree k − 1 and n vertices are of degree n. n62 + 3n + nk2 − 3nk Therefore, by using Lemma 2.3, G has size . There are four types 2 of edges in G based on degrees of end vertices of each edge. The ﬁrst edge partitions n(n − 1) E (G), contains edges uv, where d = d = n. The second edge partitions 1 2 u v E2(G), contains n edges uv, where du = k −1,dv = n. The third edge partitions E3(G), contains (k − 2)n edges uv, where du = k − 1,dv = k − 2 and the forth edge partitions nk2 + 6n − 5kn E (G), contains edges uv, where d = d = k −2. We take k −1 ≤ n, 4 2 u v from Deﬁnition 2.1, of the M-polynomial of G, we have

i j M(G; x,y) = mij x y ≤ i j n n k−1 n = mnnx y + m(k−1)nx y ≤ − ≤ m n k 1 n k−1 k−2 k−2 k−2 + m(k−1)(k−2)x y + m(k−2)(k−2)x y k−1≤k−2 k−2≤k−2 Some Computational Aspects for the line graph of Banana Tree Graph 7 n n k−1 n = mnnx y + m(k−1)nx y ∈ ∈ uv E1(G) uv E2(G) k−1 k−2 k−2 k−2 + m(k−1)(k−2)x y + m(k−2)(k−2)x y

uv∈E3(G) uv∈E4(G) n n k−1 n k−1 k−2 =|E1(G)|x y +|E2(G)|m(k−1)nx y +|E3(G)|x y k−2 k−2 +|E4(G)|x y n(n − 1) − − − = xnyn + nxk 1yn + ((k − 2)n)xk 1yk 2 2 nk2 + 6n − 5kn + xk−2yk−2. 2

Figure 3: Plot of M-polynomial for the line graph of Banana graph

Next we compute some degree-based topological indices of the line graph of banana tree from this M-polynomial.

Corollary 3.2. Let G be the line graph of the Banana tree. Then

3 2 2 1. M1(G) = n(k − 5k + (n + 10)k + n − 2n − 7). 1 2. M (G) = (k4 − 7k3 + 20k2 + (2n − 28)k + n3 − n2 − 2n + 16)n. 2 2 (n2 + n − 1)k2 + (−2n2 − n + 3)k − n2 − 2n − 2 3. mM (G) = . 2 2(k − 2)(k − 1)n 8 M. S. Ahmad, et al.

1 + 1 + + 4. R (G) = n2α 1(n−1)+ n(k −2)(k −3)(k −2)2α +((k −2)α 1n+nα 1)(k − α 2 2 1)α. 1 n(n − 1) n n(k − 2) n(k2 − 5k + 6) 5. RR (G) = + + + . α 2 n2α (k − 1)αnα (k − 1)α(k − 2)2α 2(k − 2)2α 1 + k3n + (−4n + 1)k2 + (n2 + 4n − 2)k 6. SSD(G) = . (k − 1) n − 1 n n(k − 2) n(k2 − 5k + 6) 7. H (G) = + + + . 16 n + k − 1 2k − 3 4k − 8 1 2(k − 1)n n(k − 2)2(k − 1) n(k − 2)3(k − 3) 8. I (G) = n2(n − 1) + + + . 4 n + k − 1 2k − 3 4k − 8 n7 (k − 1)7n4 n(k − 2)4(k − 1)3 n(k − 2)7 9. A(G) = + + + . 16(n − 1)2 (n + k − 3)3 (2k − 5)3 16(k − 3)2

Proof. Let M(G; x,y) = f(x,y) n(n − 1) − − − = xnyn + nxk 1yn + ((k − 2)n)xk 1yk 2 2 nk2 + 6n − 5kn + xk−2yk−2. 2 Then 2 n (n − 1) − D f(x,y) = xnyn + n(k − 1)xk 1yn x 2 2 − − nk + 6n − 5kn − − + ((k − 1)(k − 2)n)xk 1yk 2 + (k − 2) xk 2yk 2, 2 2 n (n − 1) − − − D f(x,y) = xnyn + n2xk 1yn + ((k − 2)2n)xk 1yk 2 y 2 2 nk + 6n − 5kn − − + (k − 2) xk 2yk 2, 2 3 n (n − 1) − − − D D f(x,y) = xnyn + (k − 1)n2xk 1yn + ((k − 1)(k − 2)2n)xk 1yk 2 y x 2 2 nk + 6n − 5kn − − + (k − 2)2 xk 2yk 2, 2 n(n − 1) S (f (x, y)) = xnyn + xk−1yn + nxk−1yk−2 y 2n nk2 + 6n − 5kn + xk−2yk−2, 2 Some Computational Aspects for the line graph of Banana Tree Graph 9

(n − 1) 1 n S S (f (x, y)) = xnyn + xk−1yn + xk−1yk−2 x y 2n k − 1 k − 1 nk2 + 6n − 5kn + xk−2yk−2, 2(k − 2)2 α+1 n (n − 1) + − + − − Dα(f (x, y)) = xnyn + nα 1xk 1yn + ((k − 2)α 1n)xk 1yk 2 y 2 2 nk + 6n − 5kn − − + (k − 2)α xk 2yk 2, 2 2α+1 n (n − 1) + − DαDα(f (x, y)) = xnyn + (k − 1)αnα 1xk 1yn x y 2 + − − + ((k − 1)α(k − 2)α 1n)xk 1yk 2 2 nk + 6n − 5kn − − + (k − 2)2α xk 2yk 2, 2 n(n − 1) n (k − 2)n Sα(f (x, y)) = xnyn + xk−1yn + xk−1yk−2 y 2nα nα (k − 2)α nk2 + 6n − 5kn + xk−2yk−2, 2(k − 2)α n(n − 1) n SαSα(f (x, y)) = xnyn + xk−1yn x y 2n2α (k − 1)αnα (k − 2)n nk2 + 6n − 5kn + xk−1yk−2 + xk−2yk−2, (k − 1)α(k − 2)α 2(k − 2)2α

n(n − 1) − − − S D (f (x, y)) = xnyn + (k − 1)xk 1yn + (k − 1)nxk 1yk 2 y x 2 nk2 + 6n − 5kn + xk−2yk−2, 2 n(n − 1) n2 (k − 2)2n S D (f (x, y)) = xnyn + xk−1yn + xk−1yk−2 x y 2 k − 1 k − 1 nk2 + 6n − 5kn + xk−2yk−2, 2 n(n − 1) + − − Jf (x, y) = x2n + nxn k 1 + (k − 2)nx2k 3 2 nk2 + 6n − 5kn + x2k−4, 2 (n − 1) n (k − 2)n S Jf (x, y) = x2n + xn+k−1 + x2k−3 x 4 n + k − 1 2k − 3 nk2 + 6n − 5kn + x2k−4, 4k − 8 10 M. S. Ahmad, et al.

3 n (n − 1) + − − JD D f(x,y) = x2n + (k − 1)n2xn k 1 + (k − 1)(k − 2)2nx2k 3 x y 2 2 nk + 6n − 5kn − + (k − 2)2 x2k 4, 2 n2(n − 1) (k − 1)n2 (k − 1)(k − 2)2n S JD D f(x,y) = x2n + xn+k−1 + x2k−3 x x y 4 n + k − 1 2k − 3 2 nk + 6n − 5kn − + (k − 2)2 x2k 4, 4k − 8 4 n (n − 1) − − − D3f(x,y) = xnyn + n4xk 1yn + (k − 2)4nxk 1yk 2 y 2 2 nk + 6n − 5kn − − + (k − 2)4 xk 2yk 2, 2 7 n (n − 1) − − − D3D3f(x,y) = xnyn + (k − 1)7n4xk 1yn + (k − 1)3(k − 2)4nxk 1yk 2 x y 2 2 nk + 6n − 5kn − − + (k − 2)6 xk 2yk 2, 2 7 n (n − 1) + − − JD3D3f(x,y) = x2n + (k − 1)7n4xn k 1 + (k − 1)3(k − 2)4nx2k 3 x y 2 2 nk + 6n − 5kn − + (k − 2)6 x2k 4, 2 7 − 3 3 n (n 1) 2n−2 7 4 n+k−3 Q− JD D f(x,y) = x + (k − 1) n x 2 x y 2 2 − nk + 6n − 5kn − + (k − 1)3(k − 2)4nx2k 5 + (k − 2)6 x2k 6, 2 7 − − 7 4 3 3 3 n (n 1) 2n−2 (k 1) n n+k−3 S Q− JD D f(x,y) = x + x x 2 x y 2(2n − 2)3 (n + k − 3)3 3 4 2 (k − 1) (k − 2) n − nk + 6n − 5kn − + x2k 5 + (k − 2)6 x2k 6. (2k − 5)3 2(2k − 6)3 Using Table 1, we have the following graphs of different indices.

1. M1(G) = (Dx + Dy)f (x, y)|x=y=1 = n(k3 − 5k2 + (n + 10)k + n2 − 2n − 7).

2. M2(G) = DyDx(f (x, y))|x=y=1 1 = (k4 − 7k3 + 20k2 + (2n − 28)k + n3 − n2 − 2n + 16)n. 2 Some Computational Aspects for the line graph of Banana Tree Graph 11

Figure 4: Plot for the ﬁrst Zagreb index for the line graph of Banana tree graph.

Figure 5: Plot for the ﬁrst Zagreb index for the line graph of Banana tree graph for k = 1.

3. m M2(G) = SxSy(f (x, y))|x=y=1 (n2 + n − 1)k2 + (−2n2 − n + 3)k − n2 − 2n − 2 = . 2(k − 2)(k − 1)n 12 M. S. Ahmad, et al.

Figure 6: Plot for the ﬁrst Zagreb index for the line graph of Banana tree graph for n = 1.

Figure 7: Plot for the second Zagreb index for the line graph of Banana tree graph.

4. = α α | Rα(G) Dx Dy (f (x, y)) x=y=1 1 + 1 = n2α 1(n − 1) + n(k − 2)(k − 3)(k − 2)2α 2 2 + + + ((k − 2)α 1n + nα 1(k − 1)α. Some Computational Aspects for the line graph of Banana Tree Graph 13

Figure 8: Plot for the second Zagreb index for the line graph of Banana tree graph for k = 1.

Figure 9: Plot for the second Zagreb index for the line graph of Banana tree graph for n = 1.

5. = α α | RRα(G) Sx Sy (f (x, y)) x=y=1 1 n(n − 1) n n(k − 2) n(k2 − 5k + 6) = + + + . 2 n2α (k − 1)αnα (k − 1)α(k − 2)α 2(k − 2)2α 14 M. S. Ahmad, et al.

Figure 10: Plot for the modiﬁed second Zagreb index for the line graph of Banana tree graph.

Figure 11: Plot for the modiﬁed second Zagreb index for the line graph of Banana tree graph for k = 3.

6. SSD(G)(SyDx + SxDy)(f (x, y))|x=y=1 1 + k3n + (−4n + 1)k2 + (n2 + 4n − 2)k = . (k − 1) Some Computational Aspects for the line graph of Banana Tree Graph 15

Figure 12: Plot for the modiﬁed second Zagreb index for the line graph of Banana tree graph for n = 1.

Figure 13: Plot for the generalized Randi´cindex for the line graph of Banana tree graph 1 for α = . 2

7. H (G) = 2SxJ(f(x,y))x=1 n − 1 n n(k − 2) n(k2 − 5k + 6) = + + + . 16 n + k − 1 2k − 3 4k − 8 16 M. S. Ahmad, et al.

Figure 14: Plot for the generalized Randi´cindex for the line graph of Banana tree graph 1 for k = 4 and α = . 2

Figure 15: Plot for the generalized Randi´cindex for the line graph of Banana tree graph 1 for n = 1 and α = . 2

8. I (G) = SxJDxDy(f (x, y))x=1 1 2(k − 1)n n(k − 2)2(k − 1) = n2(n − 1) + + 4 n + k − 1 2k − 3 n(k − 2)3(k − 3) + . 4k − 8 Some Computational Aspects for the line graph of Banana Tree Graph 17

Figure 16: Plot for the inverse Randi´cindex for the line graph of Banana tree graph for 1 α = . 2

Figure 17: Plot for the inverse Randi´cindex for the line graph of Banana tree graph for 1 k = 3 and α = . 2 18 M. S. Ahmad, et al.

Figure 18: Plot for the inverse Randi´cindex for the line graph of Banana tree graph for 1 n = 1 and α = . 2

Figure 19: Plot for the symmetric division index for the line graph of Banana tree graph. Some Computational Aspects for the line graph of Banana Tree Graph 19

Figure 20: Plot for the symmetric division index for the line graph of Banana tree graph for k = 2.

Figure 21: Plot for the symmetric division index for the line graph of Banana tree graph for n = 1. 20 M. S. Ahmad, et al.

Figure 22: Plot for the harmonic index for the line graph of Banana tree graph.

Figure 23: Plot for the harmonic index for the line graph of Banana tree graph for k = 1. Some Computational Aspects for the line graph of Banana Tree Graph 21

Figure 24: Plot for the harmonic index for the line graph of Banana tree graph for n = 1.

Figure 25: Plot for the inverse sum index for the line graph of Banana tree graph. 22 M. S. Ahmad, et al.

Figure 26: Plot for the inverse sum index for the line graph of Banana tree graph for k = 4.

Figure 27: Plot for the inverse sum index for the line graph of Banana tree graph for n = 1. Some Computational Aspects for the line graph of Banana Tree Graph 23

Figure 28: Plot for the augmented Zagreb index for the line graph of Banana tree graph.

Figure 29: Plot for the augmented Zagreb index for the line graph of Banana tree graph for k = 4. 24 M. S. Ahmad, et al.

9. = 3 3 3 A(G) Sx Q−2JDxDy(f (x, y)) 7 − 7 4 − 4 − 3 = n + (k 1) n + n(k 2) (k 1) 16(n − 1)2 (n + k − 3)3 (2k − 5)3 n(k − 2)7 + . 16(k − 3)2

Figure 30: Plot for the augmented Zagreb index for the line graph of Banana tree graph for n = 2.

4. Conclusions In this article we compute many topological indices for line graph of banana tree. At ﬁrst we give general closed forms of M-polynomial of this graph and then recover many degree-based topological indices out of it. These results can play a vital rule in preparation of new drugs.

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