M-polynomials and Degree based Topological Indices for the Line Graph of Firecracker Graph

Transcription

1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13, Number 6 (017), pp Research India Publications M-polynomials and Degree based Topological Indices for the Line Graph of Firecracker Graph 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 Kang 1 Department of Mathematics and RINS, Gyeongsang National University, Jinju 588, Korea. Chahn Yong Jung Department of Business Administration, Gyeongsang National University Jinju 588, 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 forms of the M-polynomials for the line graph of Firecracker graph. From the M-polynomial, we recover some degree-based topological indices. AMS subject classification: 05C1, 05C90. Keywords: M-polynomial, topological index, line graph. 1 Corresponding author.

2 M. S. Ahmad, W. Nazeer, S. M. Kang and C. Y. Jung 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. In a chemical graph, the degree of any vertex is at most 4. The distance between two vertices u and v is denoted as d(u, v) = d G (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 d v (G) or, if misunderstanding is not possible, simply by d v. 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 nanomaterial. In these studies, some Physico-chemical properties and topological indices are used to predict bioactivity of the chemical compounds see [5, 9, 6, 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 015, plays the same role in determining the closed form of many degree-based topological indices [1, 31, 3, 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 Firecracker graph by using the M-polynomial. Some of these topological indices were calculated directly in [40].. Basic definitions and literature review Here we give some basic definitions and literature review. Definition.1. The M-polynomial of G is defined as: M(G,x,y) = m ij (G)x i y j, δ i j where δ = min{d v : v V (G)}, = max{d v : v V (G)}, and m ij (G) the number of edges vu E(G) such that {d v,d u }={i, j}. Weiner [44] in 1947 approximated the boiling point of alkanes as αw(g)+βp 3 +γ, where α, β and γ are empirical constants, W(G) is the Weiner index and P 3 is the number

3 M-polynomials and Degree based Topological Indices 3 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, 1]. Randić index, [36] denoted by R 1/ (G) and introduced by Milan Randić in 1975, is also one of the oldest topological indices. The Randić index is defined as R 1/ (G) = 1. du d v uv E(G) In 1998, working independently, Bollobas and Erdos [4] and Amic et al. [] proposed the generalized Randić index and has been studied extensively by both chemists and mathematicians [3] and many mathematical properties of this index have been discussed in [6]. For a detailed survey we refer the book [7]. The general Randić index is defined as: R α (G) = 1, and the inverse (d u d v ) α uv E(G) Randić index is defined as RR α (G) = (d u d v ) α. Obviously, R 1/ (G) is the uv E(G) particular case of R α (G) when α = 1. The Randić index is a most popular, most often applied and most studied index among all other topological indices. Many papers and books such as [4, 5, 7] are written on this topological index. Randić himself wrote two reviews on his Randić index [37, 38] and there are three more reviews on it, see [0, 8, 9]. The suitability of the Randić index 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 [] introduced first Zagreb index and second Zagreb index, which are defined as: M 1 (G) = (d u + d v ) and M (G) = (d u d v ), uv E(G) uv E(G) respectively. For details about these indices we refer [8, 19, 35, 41, 4] to the readers. Both the first Zagreb index and the second Zagreb index give greater weights to the inner vertices and edges, and smaller weights to the outer vertices and edges which oppose intuitive reasoning [30]. For a simple connected graph G, the second modified Zagreb index is defined as: m M (G) = uv E(G) 1 d(u)d(v). 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

4 4 M. S. Ahmad, W. Nazeer, S. M. Kang and C. Y. Jung symmetric division index of a connected graph G, is defined as: SDD(G) = { min(du,d v ) max(d u,d v ) + max(d } u,d v ). min(d u,d v ) uv E(G) Another variant of Randić index is the harmonic index defined as H (G) =. d u + d v 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, particularly, a simple and elegant structure. The inverse sum index is defined as: I (G) = uv E(G) d u d v d u + d v. The augmented Zagreb index of G proposed by Furtula et al. [16] is defined as A(G) = { } d u d 3 v. d u + d v uv E(G) This graph invariant has proven to be a valuable predictive index in the study of heat formation in octanes and heptanes (see [16]), whose prediction power is better than atom-bond connectivity index (please refer to [7, 13, 1] 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 relates some well-known degree-based topological indices with the M-polynomial [10]. Table 1. Derivation of topological indices topological indices first Zagreb index second Zagrab index modified second Zagrab index Randić index inverse Randić index symmetric division index harmonic index inverse sum index Augmented Zagreb Index derivation from M(G; x,y) (D x + D y )(M(G; x,y)) x=y=1 (D x D y )(M(G; x,y)) x=y=1 (S x S y )(M(G; x,y)) x=y=1 (Dx α Dα y )(M(G; x,y)) x=y=1 (Sx α Sα y )(M(G; x,y)) x=y=1 (D x S y + S x D y )(M(G; x,y)) x=y=1 S x J(M(G; x,y)) x=y=1 S x JD x D y (M(G; x,y)) x=y=1 Sx 3 Q JDx 3 D3 y (M(G; x,y)) x=y=1

5 M-polynomials and Degree based Topological Indices 5 In Table 1, D x = x (f(x,y)), D y = y (f(x,y)), x y x f(t,y) y f(t,y) S x = dt, S y = dt, 0 t 0 t J(f(x,y)) = f(x,x), Q α (f (x, y)) = x α f(x,y). The following lemmas [19] will be helpful for our results. Lemma.. Let G be a graph with u, v V (G) and e = uv E(G). Then de = du + dv. Lemma.3. Let G be a graph of order p and size q. Then the line graph L(G) of G is a graph of order p and size 1 M 1(G) q. 3. Results and discussions In this section, we compute topological indices of the line graph of Firecracker graph. The Firecracker graph F n,k is the graph obtained by the concatenation of nk-stars by linking one leaf from each. The F n,k has order nk and size nk 1.F 4,7 is shown in the Figure 1. Figure 1: The Firecracker graph F 4,7 Theorem 3.1. Let G be the line graph of Firecracker graph. Then the M-polynomial of G is M(G; x,y) = x 3 y 4 + x 3 y k + x 3 y k 1 + (n 4)x 4 y 4 + (n 6)x 4 y k + (k )x k y k 1 + nk 5nk + 6n x k y k + (n )(k )x k y k. Proof. The graph G for n = 4 and k = 7 is shown in Fig.. By using Lemma., it is easy to see that the order of G is nk 1 out of which vertices are of degree 3,

6 6 M. S. Ahmad, W. Nazeer, S. M. Kang and C. Y. Jung vertices are of degree k 1,n 3 vertices are of degree 4, n(k ) vertices are of degree k, and n vertices are of degree k. Therefore, by using Lemma.3, G has size nk 3nk + 8n 8. We partition the size of G into edges of the type E(du,dv) where uv is an edge. Figure : The line graph of Firecracker graph F 4,7 In G, there are eight types of edges based on degrees of end vertices of each edge. The first edge partitions E 1 (G), contains edges uv, where d u = 3,d v = 4. The second edge partitions E (G), contains edges uv, where d u = 3, d v = k. The third edge partitions E 3 (G), contains edges uv, where d u = 3,d v = k 1. The forth edge partitions E 4 (G), contains n 4 edges uv, where d u = d v = 4. The fifth edge partitions E 5 (G), contains n 6 edges uv, where d u = 4, d v = k. The sixth edge partitions E 6 (G), contains (k ) edges uv, where d u = k 1, d v = k. The seventh edge partitions E 7 (G), contains nk 5nk + 6n edges uv, where d u = d v = k and the eighth edge partitions E 8 (G), contains (n )(k ) edges uv, where d u = k, d v = k. From definition 1, of the M-polynomial of G, wehave M(G; x,y) = i j m ij x i y j = m 34 x 3 y 4 + m 3k x 3 y k + m 3k x 3 y k k 3 k 1 + m 44 x 4 y 4 + m 4k x 4 y k + m k k 1 x k y k k k k 1 + m k k x k y k + m k k x k y k = k k uv E 1 (G) + uv E 4 (G) + uv E 7 (G) m 34 x 3 y 4 + m 44 x 4 y 4 + uv E (G) uv E 5 (G) m k k x k y k + k k m 3k x 3 y k + m 4k x 4 y k + uv E 8 (G) uv E 3 (G) uv E 6 (G) m k k x k y k m 3k x 3 y k 1 m k k 1 x k y k 1

7 M-polynomials and Degree based Topological Indices 7 = E 1 (G) x 3 y 4 + E (G) x 3 y k + E 3 (G) x 3 y k 1 + E 4 (G) x 4 y 4 + E 5 (G) x 4 y k + E 5 (G) x 4 y k + E 6 (G) m k 1 x k y k 1 + E 7 (G) x k y k + E 8 (G) x k y k = x 3 y 4 + x 3 y k + x 3 y k 1 + (n 4)x 4 y 4 + (n 6)x 4 y k + (k )x k y k 1 + nk 5nk + 6n x k y k + (n )(k )x k y k. Next we compute some degree-based topological indices for the line graph of a Firecracker tree from this M-polynomial. Corollary 3.. Let G be the line graph of a Firecracker graph. Then 1. M 1 (G) = k 3 n 5nk + (1n 4)k + 8n 8.. M (G) = 54 4k + 8n 10nk k + 1 k4 n 7 k3 n + 11nk. 3. m M (G) = 1 (51n 54)k 4 + ( 55n + 358)k 3 + (360n 744)k + ( 60n + 408)k (k 1)(k ). 4. R α (G) = 1 α + 3 α k α + (3k 3) α + (n 4)16 α + (n 6)4 α k α +(k n 5kn + 6n)(k ) α + (k )(n )k(k ) α. 5. RR α (G) = 1 α + (3k) α + (n 4) (n 6) + (3k 3) α 16 α + (4k) α + (k 4) ((k 1)(k )) α + n(k 5k + 6) (k )(n ) (k ) α + (k(k )) α. 6. SSD(G) = 1 6 6k 4 n + ( 7n + 10)k 3 + (51n 48)k + (4n 70)k 7n k(k 1) 7. H (G) = k k (k ) n + 8 k 3 + nk + ( n 4)k n k 4

8 8 M. S. Ahmad, W. Nazeer, S. M. Kang and C. Y. Jung I (G) = k k k 6 8(n 3)k (k 1)(k ) + n k k k 3 4 n(k ) (k 3) + k(k ) (n ). k A(G) = k3 (3k 3)3 + (1 + k) 3 k (n 6)k3 n + 7 (k + 3) 3 + (k 1)3 (k ) 4 (k 5) 3 Proof. Let Then M(G; x,y) = f(x,y) + 1 (k n 5kn + 6n)(k ) 6 (k 6) k3 (k )(n ). = x 3 y 4 + x 3 y k + x 3 y k 1 + (n 4)x 4 y 4 + (n 6)x 4 y k + (k )x k y k 1 + nk 5nk + 6n x k y k + (n )(k )x k y k. D x f(x,y) = 6x 3 y 4 + 6x 3 y k + 6x 3 y k 1 + 4(n 4)x 4 y 4 + 4(n 6)x 4 y k + (k ) x k y k 1 + nk 5nk + 6n (k )x k y k + (n )(k ) x k y k, D y f(x,y) = 8x 3 y 4 + kx 3 y k + (k 1)x 4 y k 1 + 4(n 4)x 4 y 4 + k(n 6)x 4 y k + (k 1)(k )x k y k 1 + nk 5nk + 6n (k )x k y k + k(n )(k )x k y k, D y D x f(x,y) = 4x 3 y 4 + 6kx 3 y k + 6(k 1)x 4 y k (n 4)x 4 y 4 + 4k(n 6)x 4 y k + (k 1)(k ) x k y k 1 + nk 5nk + 6n (k ) x k y k + k(n )(k ) x k y k, S y (f (x, y)) = 1 x3 y 4 + k x3 y k + k 1 x3 y k (n 4)x4 y k (n 6)x4 y k (k ) + k 1 xk y k 1 + nk 5nk + 6n x k y k (k ) (n )(k ) + x k y k, k

9 M-polynomials and Degree based Topological Indices 9 S x S y (f (x, y)) = 1 8 x3 y 4 + 3k x3 y k + 3(k 1) x3 y k (n 4)x4 y k (n 6)x4 y k + k 1 xk y k 1 + nk 5nk + 6n (k ) x k y k (n ) + x k y k, k D α y (f (x, y)) = 4α x 3 y 4 + k α x 3 y k + (k 1) α x 3 y k 1 + (n 4)4 α x 4 y 4 + (n 6)k α x 4 y k + (k 1) α (k )x k y k 1 + nk 5nk + 6n (k ) α x k y k + (n )(k )k α x k y k, D α x Dα y (f (x, y)) = (1)α x 3 y 4 + (3k) α x 3 y k + (3(k 1)) α x 3 y k 1 + (n 4)(16) α x 4 y 4 + (n 6)(4k) α x 4 y k + (k 1) α (k ) α+1 x k y k 1 S α y (f (x, y)) = 4 α x3 y 4 + k α x3 y k + + nk 5nk + 6n (k ) α x k y k + (n )(k ) α+1 k α x k y k, + n 6 k α x 4 y k + (k 1) α x3 y k 1 + n 4 4 α x4 y 4 (k ) (k 1) α xk y k 1 + nk 5nk + 6n (k ) α x k y k + Sx α Sα y (f (x, y)) = 1 α x3 y 4 + (3k) α x3 y k + + n 6 (4k) α x4 y k + + nk 5nk + 6n (k ) α x k y k + (n )(k ) k α x k y k, (3(k 1)) α x3 y k 1 + n 4 (16) α x4 y 4 (k ) ((k )(k 1)) α xk y k 1 (n )(k ) (k(k )) α xk y k, S y D x (f (x, y)) = 3 x3 y k x3 y k + 6 k 1 x3 y k 1 + (n 4)x 4 y 4 + 4(n 6) x 4 y k + k (k )α k 1 xk y k 1 + nk 5nk + 6n x k y k + (n )(k ) x k y k, k

10 10 M. S. Ahmad, W. Nazeer, S. M. Kang and C. Y. Jung S x D y (f (x, y)) = 8 3 x3 y 4 + k 3 x3 y k + k 1 x3 y k 1 + (n 4)x 4 y 4 k(n 6) + x 4 y k + (k 1)x k y k nk 5nk + 6n x k y k + k(n )x k y k, Jf (x, y) = x 7 + x 3+k + x +k + (n 4)x 8 + (n 6)x 4+k + (k )x k 3 + nk 5nk + 6n x k 4 + (n )(k )x k, S x Jf (x, y) = 7 x k x3+k + + k x+k + n 4 8 x8 + n 6 4 (k ) + k 3 xk 3 + nk 5nk + 6n x k 4 + 4(k ) x 4+k (n )(k ) x k, k JD x D y f(x,y) = 4x 7 + 6kx 3+k + 6(k 1)x +k + 16(n 4)x 8 + 4k(n 6)x 4+k + (k 1)(k ) x k 3 + nk 5nk + 6n (k ) x k 4 + k(n )(k ) x k, S x JD x D y f(x,y) = 4 7 x7 + 6k 6(k + 1) 3 + k x3+k + + k x+k + (n 4)x 8 + 4k(n 6) x 4+k k (k 1)(k ) k 3 x k 3 + nk 5nk + 6n (k )x k 4 k(n )(k ) + x k, 4 k Dy 3 f(x,y) = 43 x 3 y 4 + k 3 x 3 y k + (k 1) 3 x 3 y k 1 + (n 4)4 3 x 4 y 4 + (n 6)k 3 x 4 y k + (k 1) 3 (k )x k y k 1 + nk 5nk + 6n (k ) 3 x k y k + (n )(k )k 3 x k y k, D 3 x D3 y f(x,y) = (1)3 x 7 y 4 + (3k) 3 x 3 y k + (3(k 1)) 3 x 3 y k 1 + (n 4)(16) 3 x 4 y 4 + (n 6)(4k) 3 x 4 y k + (k 1) 3 (k ) 4 x k y k 1 + nk 5nk + 6n (k ) 6 x k y k + (n )(k ) 4 k 3 x k y k, JD 3 x D3 y f(x,y) = (1)3 x 7 + (3k) 3 x 3+k + (3(k 1)) 3 x +k + (n 4)(16) 3 x 8 + (n 6)(4k) 3 x 4+k + (k 1) 3 (k ) 4 x k 3 + nk 5nk + 6n (k ) 6 x k 4 + (n )(k ) 4 k 3 x k,

11 M-polynomials and Degree based Topological Indices 11 Figure 3: Plot for the first Zagreb index for the line graph of Firecracker graph Figure 4: Plot for the first Zagreb index for the line graph of Firecracker graph for k = 1 Q JD 3 x D3 y f(x,y) = (1)3 x 5 + (3k) 3 x 1+k + (3(k 1)) 3 x k + (n 4)(16) 3 x 6 + (n 6)(4k) 3 x +k + (k 1) 3 (k ) 4 x k 5 + nk 5nk + 6n (k ) 6 x k 6 + (n )(k ) 4 k 3 x k 4,

12 1 M. S. Ahmad, W. Nazeer, S. M. Kang and C. Y. Jung Figure 5: Plot for the first Zagreb index for the line graph of Firecracker graph for n = 1 Sx 3 Q JDx 3 D3 (1)3 yf(x,y) = 5 3 x 5 + (3k)3 (1 + k) 3 x1+k + Using Table 1, + (n 4)(16)3 6 3 x 6 + (3(k 1))3 k 3 x k (n 6)(4k)3 ( + k) 3 x +k + (k 1)3 (k ) 4 (k 5) 3 x k 5 + nk 5nk + 6n (k 6) 3 (k ) 6 x k 6 + (n )(k )4 k 3 3 x k M 1 (G) = (D x + D y )f (x, y) x=y=1 M (G) = (D y D x )f (x, y) x=y=1 = k 3 n 5nk + (1n 4)k + 8n 8. = 54 4k nk k + 1 k4 n 7 k3 n + 11nk. 3. m M (G) = S x S y (f (x, y)) x=y=1 = 1 (51n 54)k 4 + ( 55n + 358)k 3 + (360n 744)k + V 48 (k 1)k(k ),

13 M-polynomials and Degree based Topological Indices 13 Figure 6: Plot for the second Zagreb index for the line graph of Firecracker graph Figure 7: Plot for the second Zagreb index for the line graph of Firecracker graph for k = 1

14 14 M. S. Ahmad, W. Nazeer, S. M. Kang and C. Y. Jung Figure 8: Plot for the second Zagreb index for the line graph of Firecracker graph for n = 1 Figure 9: Plot for the modified second Zagreb index for the line graph of Firecracker graph

15 M-polynomials and Degree based Topological Indices 15 Figure 10: Plot for the modified second Zagreb index for the line graph of Firecracker graph for k = 4 Figure 11: Plot for the modified second Zagreb index for the line graph of Firecracker graph for n = 1

16 16 M. S. Ahmad, W. Nazeer, S. M. Kang and C. Y. Jung Figure 1: Plot for the generalized Randić index for the line graph of Firecracker graph for α = where V = ( 60n + 408)k 96n R α (G) = Dx α Dα y (f (x, y)) x=y=1 = 1 α + 3 α k α + (3k 3) α + (n 4)16 α + (n 6)4 α k α + (k )(k 1)(k ) α + (k n 5kn + 6n)(k ) α + (k )(n )k(k ) α. RR α (G) = Sx α Sα y (f (x, y)) x=y=1 = 1 α + (3k) α + (3(k 1)) α + n 4 (16) α + n 6 (4k) α k 4 + ((k )(k 1)) α + n(k 5k + 6) (k )(n ) (k ) α + (k(k )) α. SSD(G)(S y D x + S x D y )(f (x, y)) x=y=1 = 1 6k 4 n + ( 7n + 10)k 3 + (51n 48)k + (4n 70)k 7n k(k 1)

17 M-polynomials and Degree based Topological Indices 17 Figure 13: Plot for the generalized Randić index for the line graph of Firecracker graph for k = 1 and α = 1 Figure 14: Plot for the generalized Randić index for the line graph of Firecracker graph for n = 1 and α = 1

18 18 M. S. Ahmad, W. Nazeer, S. M. Kang and C. Y. Jung Figure 15: Plot for the inverse Randić index for the line graph of Firecracker graph for α = 1 Figure 16: Plot for the inverse Randić index for the line graph of Firecracker graph for k = 3 and α = 1

19 M-polynomials and Degree based Topological Indices 19 Figure 17: Plot for the inverse Randić index for the line graph of Firecracker graph for n = 1 and α = 1 Figure 18: Plot for the symmetric division index for the line graph of Firecracker graph

20 0 M. S. Ahmad, W. Nazeer, S. M. Kang and C. Y. Jung Figure 19: Plot for the symmetric division index for the line graph of Firecracker graph for k = 1 Figure 0: Plot for the symmetric division index for the line graph of Firecracker graph for n = 1

21 M-polynomials and Degree based Topological Indices 1 Figure 1: Plot for the harmonic index for the line graph of Firecracker graph 7. H (G) = S x J(f(x,y)) x=1 = k k (k ) n + 8 k 3 + nk + ( n 4)k n k 4 8. I (G) = S x JD x D y (f (x, y)) x=1 = k k k 6 8(n 3)k (k 1)(k ) + n + + k k k n(k ) (k 3) + k(k ) (n ). k 9. A(G) = Sx 3 Q JDx 3 D3 y (f (x, y)) = k3 (3k 3)3 + (1 + k) 3 k (n 6)k3 n + 7 (k + 3) 3 + (k 1)3 (k ) 4 (k 5) (k n 5kn + 6n)(k ) 6 (k 6) k3 (k )(n ).

22 M. S. Ahmad, W. Nazeer, S. M. Kang and C. Y. Jung Figure : Plot for the harmonic index for the line graph of Firecracker graph for k = Figure 3: Plot for the harmonic index for the line graph of Firecracker graph for n = 1

23 M-polynomials and Degree based Topological Indices 3 Figure 4: Plot for the inverse sum index for the line graph of Firecracker graph Figure 5: Plot for the inverse sum index for the line graph of Firecracker graph for k = 1

24 4 M. S. Ahmad, W. Nazeer, S. M. Kang and C. Y. Jung Figure 6: Plot for the inverse sum index for the line graph of Firecracker graph for n = 1 Figure 7: Plot for the augmented Zagreb index for the line graph of Firecracker graph

25 M-polynomials and Degree based Topological Indices 5 Figure 8: Plot for the augmented Zagreb index for the line graph of Firecracker graph for k = 1 Figure 9: Plot for the augmented Zagreb index for the line graph of Firecracker graph for n = 1

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