ABOUT THE CERTAIN TOPOLOGICAL INDICES OF THE LINE GRAPH OF V-PANTACENIC NANOTUBEHTML Full Text
ABOUT THE CERTAIN TOPOLOGICAL INDICES OF THE LINE GRAPH OF V-PANTACENIC NANOTUBE
Yingying Gao 1, Muhammad Faisal Nadeem 2, Sohail Zafar 3, Zohaib Zahid 4, Mohamad Nazri Husin 5 and Mohammad Reza Farahani 6
College of Pharmacy and Biological Engineering 1, Chengdu University, Chengdu, 610106, China.
Department of Mathematics 2, Comsat Institute of Information Technology, Lahore, Pakistan.
School of Sciences 3, 4, University of Management and Technology (UMT), Lah.
School of Informatics and Apply Mathematics 5, Universiti Malaysia Terengganu, Kuala Terengganu, 21030, Malaysia.
Department of Applied Mathematics of Iran University of Science and Technology (IUST) 6, Narmak, Tehran 16844, Iran.
ABSTRACT: A topological index is a numeric quantity associated with a graph which characterizes the topology of the graph and is invariant under graph automorphism. Topological indices such Randić, atom-bond connectivity (ABC) and geometric (GA) indices are used to predict the bioactivity of different chemical compounds. Recently, the edge version of atom-bond connectivity and geometric arithmetic indices of graph G are introduced based on the degree of an edge of line graph of G. In this paper, the closed formulas of edge version of atom-bond connectivity and geometric-arithmetic indices for V- Pantacenic nanotube are computed.
Atom-bond connectivity index, Geometric-arithmetic index, Line graph, Phenylenes
INTRODUCTION: A graph is a collection of points and lines connecting them. The points and lines of a graph are also called vertices and edges respectively. If e is and edge of G, connecting the vertices u and v, then we write e = uv and say “u and v are adjacent”. A connected graph such that, there are is a path between all pairs of vertices. The distance d (u,v) between two vertices u and v is the length of the shortest path between u and v in G. A simple graph is an un-weighted, undirected graph without loops and multiple edges.
A single number that can be used to characterize some property of the graph is called a topological index for the graph. Obviously, the number of vertices and the number of edges are topological indices. The Wiener index was the first graph invariant reported (distance based) topological index and is defined as a half sum of the distances between all the pairs of vertices in a graph 1.
Also, the edge version of Wiener index based on distance between edges was introduced by Iranmanesh et al., 2. The degree of a vertex v is the number of vertices joining to v. Also, the degree of an edge e=uv∈E(G). is the number of its adjacent vertices in V (L (G), where the line graph L (G) of a graph G is defined to be the graph whose vertices are the edges of G, with two vertices being adjacent if the corresponding edges share a vertex in G. Estrade et al., 3 proposed a topological index named the atom-bond connectivity index (shortly ABC as
where du (or dv) denotes the degree the vertex 𝑢 (or 𝑣). the reader can find some information on atom-bond connectivity index in 4-10. In 11, Farahani introduced the edge version of atom-bond connectivity index based on the end vertex degree de and df of edges 𝑒 and 𝑓 in a line graph of 𝐺 as follows:
where 𝑑e(𝐿(𝐺)) = 𝑑e denotes the degree of the edge 𝑒 in 𝐺 (see also 12).
One of the most important topological indices is well-known branching index introduced by Randić 13 which is defined as the sum of certain bond contributions calculated from the vertex degree of the hydrogen suppressed molecular graphs.
Motivated by the definition of Randić connectivity index based on the end-vertex degrees of edges in a graph connected 𝐺 with the vertex set 𝑉(𝐺) and the edge set 𝐸(𝐺) 14, 15, Vukicevic and Furtula 16 proposed a topological index named the geometric-arithmetic index (shortly 𝐺𝐴) as
where 𝑑u(𝐺)= 𝑑e denotes the degree of the vertex 𝑢 in 𝐺. The reader can find more information’s on geometric-arithmetic index in 16-19.
In 19, the edge version of geometric arithmetic index was introduced based on end-vertex degrees of edges in a line graph of 𝐺 which is a graph such that each vertex of 𝐿(𝐺) represents an edge of 𝐺; and two vertices of 𝐿(𝐺) are adjacent if and only if their corresponding edges share a common endpoint in 𝐺, as follows
where 𝑑e(𝐿(𝐺)) = 𝑑e denotes the degree of the edge 𝑒 in 𝐺.
The topological indices of H-Pantacenic nanotubes were studied recently in 20-24. With the same motivation, the aim of this note is to compute a closed formula for the 𝐴𝐵𝐶𝑒 and 𝐺𝐴𝑒 indices of V-Pantacenic nanotube.
Topological Indices of the Line Graph of V-Pantacenic Nanotube: The V-Pantacenic nanotube F[2,5] and its line graph are shown in Fig. 1 and 2 respectively.
Theorem 1: Let G = F [p,q] be a graph of V-Pantacenic nanotube with 22pq vertices and 33pq-5p edges. Then
Proof: In L(G),there are 33pq-5pvertices. It is easily seen from Fig. 2 and Lemma 1 that | V3 (L(G)) | =20p and | V4 (L(G) ) | =33pq-25p. By using Lemma 2, we get|E(L(G) ) | =66pq-20p.
FIG. 1: THE V-PANTACENIC NANOTUBE F [2, 5]
The edge set E(L(G)) divides into three edge partitions based on degrees of the end vertices, i.e. E(L(G))=E1(L(G))∪E2(L(G))∪E3(L(G)). The edge partition E1(L(G)) contains 18pedges uv, where du=dv=3, the edge partition E2(L(G)) contains 20p edges uv, where du=3 and dv=4 and the edge partition E3(L(G)) contains 66pq-58p edges uv, where du=dv=4.
After simplification we get,
Similarly one can find the expression of GA<s
Y. Gao, M. F. Nadeem, S. Zafar, Z. Zahid, M. N. Husin and M. R. Farahani
Department of Applied Mathematics, Iran University of Science and Technology (IUST), Narmak, Tehran, Iran.
12 April, 2017
16 June, 2017
29 June, 2017
01 December, 2017