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 Some Computational Aspects for the Line Graph of Banana Tree 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 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 in- dices 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 polyno- mial (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]. Randi´cindex, [36] denoted by R−1/2(G) and introduced by Milan Randi´cin 1975, is also one of the oldest topological indices. The Randi´cindex 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 Randi´cindex 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 Randi´cindex is defined as: Rα(G) α , and the inverse (dudv) uv∈E(G) α Randi´cindex 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 Randi´cindex 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. Randi´chimself wrote two reviews on his Randi´cindex [37, 38] and there are three more reviews on it, see [20, 28, 29]. The suitability of the Randi´cindex 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 ; Randi´cindex (Dx Dy )(M(G x,y))x=y=1 α α ; inverse Randi´cindex (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.