Some KV Indices of Certain Dendrimers

In this paper, we define the modified first and second KV indices, F-KV and F1-KV indices, hyper F-KV index and augmented KV index of a graph and compute exact formulas for POPAM and tetrathiafulvalene dendrimers. Furthermore, we determine the F-KV, hyper F-KV and augmented KV polynomials of POPAM dendrimers and tetrathiafulvalene dendrimers.


Introduction
A molecular graph is a simple graph such that its vertices correspond to the atoms and edges to the bonds.Chemical Graph Theory is a branch of Mathematical Chemistry which has an important effect on the development of Chemical Sciences.Numerous topological indices have been considered in Theoretical Chemistry, especially in QSPR/QSAR study, see [1,2].

Let ( ) ( )
, be a vertex set and an edge set of a finite simple connected graph G respectively.The degree ( ) of a vertex v is the number of edges incident to v. Let denote the product of the degrees of all vertices adjacent to a vertex v.We refer to [3] for undefined term and notation.
In [4], Kulli introduced the first and second KV indices, defined as We introduce the modified first and second KV indices of a graph, defined as In [5], Furtula and Gutman proposed the F-index of a graph G, defined as The F-index was studied, for example, in [6,7,8,9,10,11].
We introduce the F 1 -KV index of a graph G, defined as (3) We define the F 1 -KV polynomial of a graph G as ( ) We define the harmonic KV index of a graph G as We propose the general harmonic KV index of a graph G and it is defined as The harmonic index was studied in [12,13,14].
We introduce the augmented KV index of a graph as follows: 71 The augmented KV index of a graph G is defined as The augmented index was studied in [15,16,17].
Considering the augmented KV index, we introduce the augmented KV polynomial of a graph G as ( ) We propose the hyper F-KV index and hyper F-KV polynomial of a graph as follows: The hyper F-KV index of a graph G is defined as The hyper F-KV polynomial of a graph G is defined as Very recently, some new KV indices have been introduced and studied such as hyper KV and square KV indices [13], connectivity KV indices [19], multiplicative connectivity KV indices [20], multiplicative KV indices and multiplicative hyper KV indices [21].In this paper, we compute the modified first and second KV indices, F-KV and hyper F-KV indices, general harmonic KV index, augmented KV index of POPAM and tetrathiafulvalene dendrimers.Also the F-KV polynomial, F 1 -KV polynomial, augmented KV polynomial of POPAM and tetrathiafulvalene dendrimers are determined.For dendrimers see [22].

Results for POPAM Dendrimers
The family of POPAM dendrimers is symbolized by [ ],   based on the degree product of neighbors of end vertices of each edge is given in Table 1.
(ii) By using equation ( 4) and Table 1 2 n POD G = By using equation ( 5) and Table 1, we deduce  6) and by using Table 1, we deduce (ii) By using equation (7) and Table 1 2. ) From equation ( 5) and Table 2, we   are given by 2 n TD (i) By using equation (6) and Table 2, we obtain       (7) and by using Table 2

2 2 POD is shown in Figure 1 . 72 Figure 1 .
Figure 1.The graph of

2 2 POD
Let G be the graph of a POPAM dendrimer [ ].

77 3 .
Results for Tetrathiafulvalene DendrimersWe consider the family of tetrathiafulvalene dendrimers.This family of dendrimers is symbolized by [ ],2 n TD where n is the steps of growth in this type of dendrimers.

Figure 2 .
Figure 2. The graph of

2 2 TD
Let G be the graph of a tetrathiafulvalene dendrimer [ ].
edge partition of G based on the degree product of neighbors of end vertices of each edge is given in Table

Theorem 9 .
11), we obtain the desired result.The augmented KV index and its polynomial of [ ] n TD 2

Table 1 .
Edge partition of