On the Sanskruti Index of Circumcoronene Series of Benzenoid

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. In such a simple molecular graph, vertices represent atoms and edges represent bonds. The Sanskruti index ( ) S G is a topological index was defined as

( ) , respectively.In chemical graphs, the vertices correspond to the atoms of the molecule, and the edges represent to the chemical bonds.Note that hydrogen atoms are often omitted.If e is an edge of G, connecting the vertices u and v, then we write e uv = and say "u and v are adjacent".A connected graph is a graph such that there is a path between all pairs of vertices.
Mathematical chemistry is a branch of theoretical chemistry for discussion and prediction of the molecular structure using mathematical methods without necessarily referring to quantum mechanics.Chemical graph theory is a branch ( ) ( ) where u d denotes G degree of vertex u.One of the important classes of connectivity indices is Sanskruti index ( ) S G defined as [5] ( ) Here our notation is standard and mainly taken from standard books of

Main Results and Discussions
In this section, we compute the Sanskruti index ( ) , see Figure 1 and Figure 2 where they are shown, also for more study and historical details of this benzenoid molecular graphs see the paper series [6]- [15].
At first, consider the circumcoronene series of benzenoid k H for all integer number 1 k ≥ .From the structure of k H (Figure 2) and references [17]- [23], one can see that the number of vertices/atoms in this benzenoid molecular , , H H H of the circumcoronene series of benzenoid [16].graph is equal to ( ) and the number of edges/bonds is equal to Because, the number of vertices/ atoms as degrees 2 and 3 are equal to 6k and ( ) and in circumco-ronene series of benzenoid molecule, there are two partitions , E E and 6 E by red, green and black color in Figure 2.
From Figure 2, one can see that the summation of degrees of vertices of molecule benzenoid k H are in four types, as follow: 2 3 5 for , and 6 for , and 3 7 for , and 3 9 for , and So, the Sanskruti index for circumcoronene series of benzenoid ( ) In such a simple molecular graph, vertices represent atoms and edges represent bonds.The Sanskruti index ( ) S G is a topological index was defined as molecular graph without directed and multiple edges and without loops, the vertex and edge sets of it are represented by ( ) How to cite this paper: Gao, Y.Y., Farahani, M.R., Sardar, M.S. and Zafar, S. (2017) On the Sanskruti Index of Circumcoronene Series of Benzenoid.Applied Mathematics, 8, 520-524.https://doi.org/10.4236/am.2017.84041 of mathematical chemistry which applies graph theory to mathematical modeling of chemical phenomena [1] [2] [3].This theory had an important effect on the development of the chemical sciences.In mathematical chemistry, numbers encoding certain structural features of organic molecules and derived from the corresponding molecular graph, are called graph invariants or more commonly topological indices.Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry.One of the best known and widely used is the connectivity index, introduced in 1975 by Milan Randić[4], who has shown this index to reflect molecular branching.
G for circumcoronene series of benzenoid.The circumcoronene series of benzenoid is family of molecular graph, which consist several copy of benzene 6 C on circumference.The first terms of this series are 1 benzene,

Figure 2 .
Figure 2. The general representation k H of the circumcoronene series of benzenoid