Eccentricity based topological indices of siloxane and POPAM dendrimers

Abstract: A massive of early drug tests indicates that there is some strong inner connections among the bio-medical and pharmacology properties of nanostar dendrimers and their molecular structures. Topological descriptors are presented as fundamentally transforming a molecular graph into a number. There exist various categories of such descriptors particularly those descriptors that based on edge and vertex distances. Topological descriptors are exercised for designing biological, physico-chemical, toxicological, pharmacologic and other characteristics of chemical compounds. In this paper, we study infinite classes of siloxane and POPAM dendrimers and derive their Zagreb eccentricity indices, eccentric-connectivity and total-eccentricity indices.


Introduction
The mathematical sciences furnish the terminology for quantitative science, and this terminology is developing in many orientations as computational science in general continues its rapid development. A timely possibility now exists to grow and strengthen the useful impacts of chemistry by enhancing the relationship among the mathematical sciences and chemistry. Computational chemistry is a outgrowth of theoretical chemistry, the natural rule of which includes the construction and distribution of a penetrating conceptual infrastructure for the chemical sciences, at the atomic and molecular levels.
The mathematical sciences have been crucial allies and have provided important tools for that rule.
Cheminformatics is a incorporation of mathematics, information science and chemistry. It interpretation quantitative structure-activity and quantitative structureproperty correlation that are beneficial to save the biological activities and characteristics of various chemical compounds, see (Devillers and Balaban, 1999). In QSAR/ QSPR investigation, physico-chemical characteristics and topological descriptors such as Wiener, Szeged, Randić, ABC and Zagreb indices are used to save the bioactivity of incompatible chemical compounds. If G presents the family of simple finite graphs then a topological descriptor Top:G l → R is a function such that for any G l ,G m ∈ G, Top(G l ) = Top(G m ) when G l and H m are both isomorphic. A graphical invariant is a quantity associated to a graph that is structural invariant.
Let G be a molecular simple and connected graph. The length of the smallest path along the vertices u and v is known as the distance among u and v. The degree of w∈V(G), recognized as d G (w), is the number of linked vertices to w in G. The eccentricity of w is the greatest distance among w and any other vertex of G.
The Wiener index, (Wiener, 1947), is investigated as half of the sum of the distances of all the pairs of vertices in G. Also a distance based topological descriptor of G is the eccentric-connectivity ξ(G) index (Sharma et al., 1997), interpreted as: and also the total eccentricity index of G presented by: For further aspects of these beneficial indices, please see: Ashrafi et al. (2009a), Ashrafi et al. (2009b), Ashrafi and Sadati (2009) Zagreb eccentricity indices are investigated by (Ghorbani and Hosseinzadeh, 2012), and are expressed as: For motivation and recent results on these topological indices related to the dendrimers, please see: Diudea et al. (2010), Malik and Farooq (2015a), Malik and Farooq (2015b), Zheng et al. (2019), and Iqbal et al. (2020).
In this paper, we consider POPAM and siloxane dendrimers, and work out their eccentricity topological indices. Also, we study the graphical comparison of eccentricity indices siloxane and POPAM dendrimers.
Dendrimers are hyper-branched macromolecule. They are being analyzed for their applications in nanotechnology, genetherapy and different areas. Each dendrimer consist of multi-functional core molecule with a dendritic wedge connected to each functional site. The core molecule without surrounding dendrons is commonly mentioned to as zero generations. Each successive iterative method along all branches forms every next generation, first generation and second generation until the terminating generation. Now a days, the topological investigation of this nanostructure has been a great interest of many researchers in chemical graph theory.

Some known results about siloxane and POPAM dendrimers
The ABC and GA indices of siloxane SD[r] and POPAM PD[r] dendrimers were studied in (Husin et al., 2015). The eccentricity based invariants of a hetrofunctional dendrimer were recently studied in (Farooq et al., 2015). Now we consider the siloxane dendrimers (SD[r]) with trifunctional core unit with r growth stages. The graphs of siloxane dendrimers for r = 1 and r = 3 are presented in Figure 1. The order and size of SD[r] are 3×2 r+2 −7 and 3×2 r+2 −8, respectively.
The POPAM dendrimer PD[r] with trifunctional core unit with r growth stages. The graphs of POPAM dendrimers for r = 1 and r = 3 are presented in Figure 2. The order and size of PD[r] are 2 r+5 −10 and 2 r+5 −11, respectively.

Main discussion and results
We start this section by computing the eccentricconnectivity index of SD 1 [r] and PD[r] depicted in Figures 1 and 2, respectively.

Theorem 3.1
The eccentric-connectivity index of the siloxane nanostar dendrimer SD[r] is:

Proof
As the structure of SD 1 [r] is symmetric, so we take only one branch of SD[r] to obtain the result as located in Figure 1. We consider a representative from a V(SD[r]), which have same eccentricity and degree. These representatives are categorized by u, v, w, u m , v m , w m for 1 ≤ m ≤ r, and are presented in Table 1, along their rate of repetition. Using Table 1, the eccentric-connectivity index of SD[r] for r ≥ 1, can be picked up as follows: 1 1 (2r+2) 4 1 (2r 1) 2 3 (2r 2) 1 3 2 (2r+2 2) 4 (2r+2 +1) 2 3 2 (2r+2 +2) After simplification, we get: This completes the proof. Theorem 2 provides eccentric-connectivity index of POPAM dendrimers PD[r].

Theorem 3.2
The eccentric-connectivity index of PD[r] is:

Proof
Similarly here we take exactly one branch of PD[r] as marked in Figure 2. We pike up one representative from the V(PD[r] that have equal eccentricity and degree. These representatives are denoted by x, y, z, a m , b m , c m , d m for 1 ≤ m ≤ r are shown in Table 2 with their rate of repetition.
Using the statistics presented in  Figure 3. It is easy to see

Representatives
Degree Eccentricity Rate of repetition

Proof
From  The comparison of first and second Zagreb-eccentricity index of siloxane nanostar dendrimer SD[r] is depicted in Figure 5. It is easy to verify that in the given domain the M 2 * (SD[r]) index of siloxane nanostar dendrimer SD[r] is more dominating.

Proof
We apply the values of In Figure 6, comparison of first Zagreb-eccentricity index of PD[r] and second Zagreb eccentricity index of PD [r] which shows that in the given domain M 1 * (PD[r]) is more dominating.
All above results are concluded in the following Table 5.

Conclusion
In this paper, we study siloxane and POPAM nanostar dendrimers and figure out their eccentric-connectivity, total-eccentricity, Zagreb-eccentricity indices and certain new formulas are obtained. Moreover we represents the comparison of the eccentric-connectivity index ξ(SD[r]) and the eccentric-connectivity index of PD[r] graphically. Also, graphically we represents the comparison of the total-eccentricity index ξ(SD[r]) and the total-eccentricity index of PD [r]. Finally we represents the comparison of Zagreb-eccentricity indices of siloxane nanostar dendrimer SD [r], and also presents the comparison of Zagreb-eccentricity indices of PD [r]. Next, we are focused to figure out certain new architectures and determine their topological descriptors which will be useful to recognize their topologies.