#### ABOUT THE CERTAIN TOPOLOGICAL INDICES OF THE LINE GRAPH OF V-PANTACENIC NANOTUBE

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**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.

Keywords: |

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 d_{u} (or d_{v}) 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 d_{e} and d_{f} of edges 𝑒 and 𝑓 in a line graph of 𝐺 as follows:

where 𝑑_{e}(𝐿(𝐺)) = 𝑑_{e} 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 𝑑_{u}(𝐺)= 𝑑_{e} 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 𝑑_{e}(𝐿(𝐺)) = 𝑑_{e} 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 | V_{3} (L(G)) | =20p and | V_{4 }(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))=E_{1}(L(G))∪E_{2}(L(G))∪E_{3}(L(G)). The edge partition E_{1}(L(G)) contains 18pedges uv, where d_{u}=d_{v}=3, the edge partition E_{2}(L(G)) contains 20p edges uv, where d_{u}=3 and d_{v}=4 and the edge partition E_{3}(L(G)) contains 66pq-58p edges uv, where d_{u}=d_{v}=4.

Then:

After simplification we get,

Similarly one can find the expression of GA<s

# Article Information

52

5349-5352

483KB

602

English

IJPSR

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.

MrFarahani88@Gmail.com

12 April, 2017

16 June, 2017

29 June, 2017

10.13040/IJPSR.0975-8232.8(12).5349-52

01 December, 2017