Molecular Irregularity Indices of Nanostar, Fullerene, and Polymer Dendrimers

Dendrimers are highly branched organic macromolecules with successive layers of branch units surrounding a central core. Some properties like toxicity, entropy, and heats of vaporization of these dendrimers can be forecasted using topological indices. *e present article is devoted to study of irregularity indices of three well-known classes of dendrimers, namely, nanostar dendrimer D [p], fullerene dendrimer NS4[p], and polymer dendrimerNS5[p], where p is the step size. We also see the relation of irregularity of these dendrimers on the step size graphically.


Introduction
Dendrimers are nanosized, gradually well-formed molecules with precise, homogeneous, and monodispersed structure consisting of tree-like arms or branches used as anticancer drug [1]. ese molecules were formed initially in 1978, by D. Tomalia and coworkers in the early 1980s and at the same time independently by George. Dendrimers might be labeled as "cascade molecules," but this term is not fixed as dendrimers [2][3][4]. Dendrimers are approximately monodisperse macromolecules consisting of symmetric branched units constructed around a small molecule or a linear polymer core [5][6][7]. Dendrimers are used in supermolecular chemistry, especially in hot-guest reactions and self-assembly processes.
ese are eminently defined artificial macromolecules, which are described by a combination of a high number of functional groups and a compact molecular structure [3]. ese materials are fresh addition in a class of macromolecular nanoscale delivery devices [8]. In Figure 1, the main three general parts of a dendrimer are elaborated. e formations of dendrimer molecules start with the central atom or group of atoms called core [5,[9][10][11][12]. ere is a debate about the exact arrangement of dendrimers, whether they are fully enlarged with high density at the surface or whether the end-groups bend back into a closely packed interior [5,11,12]. Dendrimers have several applications in different fields of medicines like anticancer drugs in biomedical fields, a transdermal drug delivery, and gene delivery and as a magnetic resonance imaging contrast agent and as a dendritic sensor [13][14][15]. e subject matter of the present article is degree-based irregularity indices of some dendrimers. We are interested in the study of irregularity determinants of some famous dendrimers, namely, nanostar dendrimer D[p], fullerene dendrimer NS 4 [p], and polymer dendrimer NS 5 [p]. It has been keenly observed that the geometry and pattern of chemical systems characterize its physical aspects [16][17][18][19][20][21][22][23][24].

Preliminaries and Notations
Let G be a simple connected graph with vertex V and edge set E, and d u be the degree of vertices u. A topological invariant is an isomorphism of the graph that preserves the topology of the graph. A graph is said to be regular if every vertex of the graph has the same degree. A topological invariant is called an irregularity index if this index vanishes for a regular graph and is nonzero for a nonregular graph. Regular graphs have been extensively investigated, particularly in mathematics. eir applications in chemical graph theory came to be known after the discovery of nanotubes and fullerenes.
Paul Erdos emphasized this in the study of irregular graphs for the first time in history in [25]. In the Second Krakow Conference on Graph eory, Erdos officially posed an open problem to " determine the extreme size of highly irregular graphs of given order" [26][27][28]. Since then, irregular graphs and the degree of irregularity have become one of the core open problems of graph theory. A graph in which each vertex has a different degree than the other vertices is known as a perfect graph. e authors of [27] demonstrated that there does not exist any perfect graph. e graphs lying in between are called quasiperfect graphs, in which all except two vertices have different degrees [29]. Irregularity of networks is discussed in [30]. Simplified ways of expressing the irregularities are irregularity indices. ese irregularity indices have been studied recently in a novel way [31][32][33]. e first such irregularity index was introduced in [32]. Most of these indices used the concept of the imbalance of an edge defined as imball uv � |du − dv| [33]. e Albertson index, AL(G), was defined by Albertson in [32] as AL(G) � UV∈E |d u − d v |. In this index, the imbalance of edges is computed. e irregularity indices IRL(G) and IRLU(G) are introduced by Vukičević and Graovac [33] as IRL(G) � Recently, Abdo et al. introduced the new term "total irregularity measure of a graph G", which is defined as IRR t (G) � 1/2 UV∈E |d u − d v | [34][35][36]. Recently, Gutman introduced the IRF(G) irregularity index of the graph G, which is described as e Randic index itself is directly related to an irregularity measure, which is described as IRA(G) � [38]. Further irregularity indices of similar nature can be traced in [37,39] in detail. ese indices are given as  [45] and number of spanning trees and normalized Laplacian of some network in [46].
In the current article, we are interested in finding the degree of irregularity of the nanostar dendrimers D[p], fullerene dendrimerNS 4 [p], and polymer dendrimerNS 5 [p]. Figures 2-4 represent molecular graphs of these three systems. e main motivation comes from the fact that graphs of the irregularity indices show close and accurate results about properties like entropy, standard enthalpy, vaporization, and acentric factors of octane isomers [39]. e molecular pattern and topology of these three dendrimers are shown in these figures. Figure 2 represents the structure of D[p] nanostar dendrimer. In Figures 3 and 4, structure of NS 4 [p] fullerene dendrimers and NS 5 [p] polymer dendrimer is shown, respectively.

Main Results
In this section, we present our main theoretical results. Theorem 1. Let D p be the nanostar dendrimer, then the irregularity indices of D p are as follows: Proof. In order to prove the above theorem, we have to consider Figure 2 along with Table 1. Now using Table 1 and the above definitions, we have   Journal of Chemistry   Journal of Chemistry 3 Journal of Chemistry Proof. In order to prove the above theorem, we have to consider Figure 3 and Table 3. Now using Table 3 and the above definitions, we have Journal of Chemistry 5    Journal of Chemistry Proof. In order to prove the above theorem, we have to consider Figure 4 and Table 5. Now, using Table 5 and the above definitions, we have

Irregularity indices
Journal of Chemistry

Graphical Analysis, Discussions, and Conclusions
In the present section, we give the graphical analysis of the irregularity measures of the above three classes of dendrimer structures. We summarize our findings in terms of graphs of irregularity indices against the step size p from 1 to 30. Figure 5 presents three different curves for the values of irregularity index IRDIF against the step size p. It can be demonstrated that irregularity of all three dendrimers tends to increase with increase in step size. It is noticeable that for p > 16, IRDIF rises sharply showing that irregularity is nonuniform.
In Figure 6, curves of irregularity index IRR have been drawn against step size p and behavior of this index is similar to IRIDF.
All irregularity indices show the similar pattern for three dendrimers, as indicated in Figures 7-9.
From the above discussion, it can be concluded that NS 5 [p] is relatively more irregular than NS 4   most regular in these three dendrimers. e dependence of irregularity of structures on step size could be a potential information career for modelling nanoscale drugs and devices.

Data Availability
No such data are associated with this article.

Conflicts of Interest
e authors declare no conflicts of interest.