On Reverse Eccentric Connectivity Index of One Tetragonal Carbon Nanocones

Research into carbon nanocones (CNC) started almost at the same time as the discovery of carbon nanotube (CNT) in 1991. In resent years, nanostructures involving carbon have been the focus of an intense research activity, which is driven to a large extent by the quest for new materials with specific applications. Ball studied the closure of (CNT) and mentioned that (CNT) could sealed by a conical cap, [1]. The official report of the discovery of isolated CNC was made in, 1994 when Ge and Sattler reported their observations of carbon cones mixed together with tubules an a flat graphite surface [2]. This are constructed from a graphene sheet by removing a 60o wedge and joining the edges produces a cone with a single pentagonal defect at the apex. If a 120o wedge is considered then a cone with a single square defect at the apex is obtained. the case of 240o wedges yields a single triangle defect at the apex [3-5].


Introduction
Research into carbon nanocones (CNC) started almost at the same time as the discovery of carbon nanotube (CNT) in 1991. In resent years, nanostructures involving carbon have been the focus of an intense research activity, which is driven to a large extent by the quest for new materials with specific applications. Ball studied the closure of (CNT) and mentioned that (CNT) could sealed by a conical cap, [1]. The official report of the discovery of isolated CNC was made in, 1994 when Ge and Sattler reported their observations of carbon cones mixed together with tubules an a flat graphite surface [2]. This are constructed from a graphene sheet by removing a 60º wedge and joining the edges produces a cone with a single pentagonal defect at the apex. If a 120º wedge is considered then a cone with a single square defect at the apex is obtained. the case of 240º wedges yields a single triangle defect at the apex [3][4][5].
Topological indices are graph invariants and are used for Quantitative Structure -Activity Relationship (QSPR) and Quantitative Structure -Property Relationship (QSPR) studies, [6][7][8]. Many topological indices have been defined and several of them have found applications as means to model physical, chemical, pharmaceutical and other properties of molecules.
A topological index of a molecular graph G is a numeric quantity related to G. The oldest nontrivial topological index is the Wiener index which was introduced by Harold Wiener. John Platt was the only person who immediately realized the importance of the Wiener's pioneering work and wrote papers analyzing and interpreting the physical meaning of the Wiener index.
We now recall some algebraic definitions that will be used in the paper. Let G be a simple molecular graph without directed and multiple edges and without loops, the vertex and edge-sets of which are represented by V(G) and E(G), respectively. The vertices in G are connected by an edge if there exists an edge uv∈ E(G) connecting the vertices u and v in G so that u,v∈ V(G). In chemical graphs, the vertices of the graph correspond to the atoms of the molecule, and the edges represent the chemical bonds. The number of vertices and edges in a graph will be defined by |V (G)| and |E (G)| respectively. In graph theory, a path of length n in a graph is a sequence of n+1 vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A vertex is external, if it lies on the boundary of the unbounded face of G, otherwise, the vertex is called internal.
is defined as the length of the shortest path between u and v in G. D(u) denotes the sum of distances between u and all other vertices of G. For a given vertex u of V(G) its eccentricity, ecc (u), is the largest distance between u and any other vertex v of G.
The maximum eccentricity over all vertices of G is called the diameter of G and denoted by Diam(G) and the minimum eccentricity among the vertices of G is called radius of G and denoted by R (G) and for any vertex u, S u is the sum of the degrees of its neighborhoods and deg G (u) denotes the degree of the vertex u.
The Wiener index [9] is one of the most used topological indices with high correlation with many physical and chemical indices of molecular compounds. The Wiener index of a molecular graph G, denoted by W(G), is defined The eccentric connectivity index of the molecular graph G, , was proposed by Sharma. V and Gosvami [4]. It is defined as The modified eccentric connectivity index (MEC) is defined as.
Recently, Ediz et al. [10] introduced a distance-based molecular structure descriptor, the reverse eccentric connectivity index defined as, In this paper by using an algebraic method, we calculate the reverse eccentric connectivity index of one tetragonal carbon nanocones.

Result and Discussion
Let . Our notation is standard and mainly taken from standard books of graph theory and the books of Trinajestic and

Abstract
Let be a molecular graph. The reverse eccentric connectivity index is defined as where ecc(u) is a largest distance between u and any other vertex v of molecular graph G and S u is the sum of the degrees of all vertices v, adjacent to vertex u. In this paper, an exact formula for the reverse eccentric connectivity index of one tetragonal carbon anocones was computed. Kumar [11,12]. In this section, the reverse eccentric connectivity index of C[n] are calculated. To do this, the following lemmas are necessary. In the following theorem, the reverse eccentric connectivity index of C[n] is computed when ( 1) n ≥ is an odd number.

Theorem:
The reverse eccentric connectivity index of C[n] is given by :