Minus F and Square F-Indices and Their Polynomials of Certain Dendrimers

We introduce the minus F-index and square F-index of a graph. In this study, we determine the minus F-index, square F-index and their polynomials of porphyrin dendrimer, propyl ether imine dendrimer, zinc porphyrin dendrimer and poly ethylene amide amine dendrimer.


Introduction
Let G be a finite, simple, connected graph with vertex set ( ) and edge set ( ).

G E
The degree ( ) of a vertex v is the number of edges incident to v. The edge connecting vertices u and v will be denoted by uv.For other definitions and notations, readers are referred to [1].
Chemical Graph Theory is a branch of Mathematical Chemistry which has an important effect on the development of Chemical Sciences.A molecular graph or chemical graph is a simple graph such that its vertices correspond to the atoms and the edges to the bonds.In Chemistry, topological indices have been found to be useful in discrimination, chemical documentation, structure property relationships, structure activity relationships and pharmaceutical drug design.There has been considerable interest in the general problem of determining topological indices, see [2,3].
The irregularity index (called as minus index [8]) was introduced by Albertson in [9] and defined as Recently, the square ve-degree index was introduced by Kulli in [10] and defined as Very recently, some square indices were introduced and studied such as square reverse index [11], square Revan index [12] square leap index [13], square KV index [14].
We now introduce the minus F-index and square F-index of a graph G as follows: The minus F-index of a graph G is defined as The square F-index of a graph G is defined as Considering the minus F and square F indices, we define the minus F and square F polynomials of a graph G as ( ) ( ) 173 In this paper, we consider the porphyrin, propyl ether imine, zinc porphyrin and poly ethylene amide amine dendrimers.Some degree based topological indices, eccentricity based topological indices of these dendrimers were studied in [15,16,17,18,19,20].In Chemical Graph Theory, graph polynomials related to molecular graph were studied in [21,22,23,24,25,26,27,28,29].Graph polynomials and topological based numbers have significant importance to collect information about properties of chemical compounds [30].In this paper, the minus F and square F indices and their polynomials of porphyrin, propyl ether imine, zine porphyrin and poly ethylene amide amine dendrimers are determined.In G, there are six types of edges based on degrees of end vertices of each edge.By calculation, the edge partition of G is given in Table 1.

Results for Porphyrin Dendrimer
Table 1.Edge partition of .
( ) By using equation ( 3) and Table 1, we have   Let .
From equation ( 4) and by using Table 1, we derive ( )

Results for Propyl Ether Imine Dendrimer PETIM
We consider the propyl ether imine dendrimer which is denoted by PETIM.This dendrimer is presented in Figure 2. In G, there are exactly three types of edges based on degrees of end vertices of each edge.Also by calculation, the edge partition of G is given in Table 2.
from equation (3) and by using Table 2, we derive ( )  2, we obtain From equation (4) and by using Table 2, we have ( )  edges.In G, there are four different types of edges based on degrees of end vertices of each edge.Also by calculation, the edge partition of G is given in Table 3.Let  .

Results for Poly Ethylene Amide Amine Dendrimer PETAA
We consider the poly ethylene amide amine dendrimer which is denoted by PETAA.This dendrimer is shown in Figure 4. edges.In G, there are four different types of edges based on degrees of end vertices of each edge.Also by calculation, the edge partition of G is given in Table 4.
From equation (1) and by using Table 4, we obtain We consider the porphyrin dendrimer which is denoted by .n n P D The porphyrin dendrimer is shown in Figure 1.

Theorem 7 .
The square F index of PETIM is By using equation(2) and Table zinc porphyrin dendrimer and it is symbolized by .n DPZ The zinc porphyrin dendrimer is depicted in Figure 3.

Theorem 11 .
The square F-index of n

Figure 4 .
Figure 4. Poly ethylene amide amine dendrimer PETAA.Let PETAA G = be a poly ethylene amide amine dendrimer.By calculation, G has 18 2 44 − × n vertices and 19 2 44 − × nedges.In G, there are four different types of edges based on degrees of end vertices of each edge.Also by calculation, the edge partition of G is given in Table4.

Table 3 .
Edge partition of n DPZ .
The square F-polynomial of PETAA is n Theorem 16.