First and Second Zagreb polynomials of VC 5 C 7 [p,q] and HC 5 C 7 [p,q] nanotubes

Let G = (V,E) be a simple connected graph. The sets of vertices and edges of G are denoted by V = V(G) and E = E(G), respectively. There exist many topological indices and connectivity indices in graph theory. The First and Second Zagreb indices were first introduced by Gutman and Trinajstić in 1972. It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical instances, and in elsewhere. In this paper, we focus on the structure of ” G = VC 5 C 7 [p,q]” and ” H = HC 5 C 7 [p,q]” nanotubes and counting First Zagreb index Zg 1 (G) = 2 ( ) v v V G d   and Second Zagreb index Zg 2 (G) =   ( ) u v e uv E G d d     of G and H, as well as First Zagreb polynomial Zg 1 (G,x ) = ( ) u v d d e uv E G x     and Second Zagreb polynomial Zg 2 (G,x) = ( ) . u v d d e uv E G x    


INTRODUCTION
Let G be a simple molecular graph with vertex and edge sets V(G) and E(G), respectively. As usual, the distance between the vertices u and v of G is denoted by d G (u,v) (or d(u,v) for short) and it is defined as the number of edges in a minimal path connecting vertices u and v [3,16,17].
In graph theory, we have many different connectivity index and topological index of arbitrary graph G. A topological index is a numeric quantity from the structural graph of a molecule which is invariant under graph automorphisms. Usage of topological indices in chemistry began in 1947 when chemist Harold Wiener developed the most widely known topological descriptor.
The Wiener index W(G) is the oldest topological indices, (based structure descriptors) [4,10,14,15,18,19], which have very chemical applications, mathematical properties and defined as follow: An important topological index introduced more than forty years ago by I. Gutman and Trinajstić is Zagreb index Zg 1 (G) (or, more precisely, the First Zagreb index, because there exists also a Second Zagreb index, Zg 2 (G) [17]). First Zagreb index Zg 1 (G) of the graph G is defined as the sum of the squares of the degrees of all vertices of G. They are defined as: where d u and d v are the degrees of u and v, respectively.
Also, we have First Zagreb polynomial Zg 1 (G,x) and Second Zagreb polynomial Zg 2 (G,x) for two above topological indices. They are defined as [1,[6][7][8][9]: The mathematical properties of these topological indices can be found in some recent papers [1][2][3][6][7][8][9]17]. In this paper, we focus on First Zagreb polynomial, Second Zagreb polynomial and their topological indices of "G = VC 5 C 7 [p,q]" and "H = HC 5 C 7 [p,q]" nanotubes. Molecular graphs "VC 5 C 7 [p,q]" and "HC 5 C 7 [p,q]" are tow family of nanotubes, such that their structure are consist of cycles with length five and seven by different compound. In other words, a C 5 C 7 net is a trivalent decoration made by alternating C 5 and C 7 . It can cover either a cylinder or a torus. For a review, historical details and further bibliography see the 3dimensional lattice of "VC 5 C 7 [p,q]" and "HC 5 C 7 [p,q]" nanotubes in Figure 1 and their 2dimensional lattice in Figure 2 and Figure 3, respectively and references [6,[11][12][13].

RESULTS AND DISCUSSION
On the other hands, to achieve our aims and counting our favorites indices of "VC 5 C 7 [p,q]" and "HC 5 C 7 [p,q]" nanotubes, we need to the following definition.    So Second Zagreb index of G is Zg 2 (VC 5 C 7 [p,q]) = 216pq +18p.

Proof.
Consider nanotubes G = VC 5 C 7 [p,q], wedenote the number of pentagons in the first row by p, in this nanotubes the four first rows of vertices and edges are repeated alternatively, we denote the number of this repetition by q. Hence the number of vertices in this nanotubes is equal to (p,q N) n = |V(VC 5 C 7 [p,q])| = 16pq + 6p. Since |V 2 | = 3p + 3p and |V 3 | = 16pq, thus e = |E(VC 5 So, we mark the edges of E5, E 6 * by red color and the edges of E6, E 9 * by black color, in Figure 2. Thus, we have the number of 6p+6p and 24pq-6p members edges of edge set E 5 (or E 6 *) and E 6 (or E 9 *) of G=VC 5  By according to the definition of First Zagreb index and Second Zagreb index, we have following equations: Here, we complete the proof of Theorem 1.

Proof.
Consider nanotubes H = HC 5 C 7 [p,q]. This nanotubes consists of heptagon and pentagon nets. We denote the number of heptagons in the first row by p. In this nanotubes the four first rows of vertices and edges are repeated alternatively, we denote the number of this repetition by q. Hence the number of vertices in this nanotubes is equal to n = |V(HC 5