1. Introduction
Nanotechnology is a modern and fast-growing field that requires the design, production, and exploitation of structures at the nanoscale. A nano-object with all three external dimensions in the nanoscale is specified as a nanoparticle. One of the most brilliant applications of nanotechnology is in the discipline of medicine [
1]. Nanomedicine performs a significant role in drug delivery, cancer treatment, and so on. Dendrimers are nano-sized, radially symmetric molecules with the very well homogeneous and monodisperse structure containing tree-like arms [
2,
3,
4,
5]. Dendrimers are recognised by unusual properties like small size, high functionality, cavities, well-defined three-dimensional structure, and globular shape. These properties make them rare candidates for using in nanotechnology and diverse biomedical applications [
6,
7]. Dendrimer structures are shaped with a fundamental atom or group of atoms named as the core, from this central structure, branches of other atoms called “dendrons” raise through various chemical reactions [
4]. Dendrimers show significantly magnified physical and chemical properties compared to usual linear polymers. At present time, dendrimers are inviting the interest of a great number of scientists because of their wide range of talented applications in various fields such as material, nanoscience, biology, medicine, physics [
8,
9,
10,
11,
12,
13,
14,
15].
In theoretical chemistry and biology, topological indices have been used for working out the information on molecules in the form of numerical coding. This relates to characterizing physico-chemical, biological, toxicologic, pharmacologic, and other properties of chemical compounds by taking advantage of molecular indices. Thousands of molecular structure descriptors have been suggested in order to characterize the physical and chemical properties of molecules. Those indices can be separated into various classes, specifically degree-based indices, distance-based indices, eigenvalue-based indices, and mixed indices. Degree-based indices can be further classified in the class of irregularity indices that measure the irregularity of the given graph. Recently, R
ti et al. [
16] showed that the graph irregularity indices are efficient in quantitative structure property relationships (QSPR) studies of molecular graphs.
Throughout this article, all graphs are finite, undirected and simple. Let
be such a graph with vertex set
and edge set
. The order of
G is the cardinality of its vertex set and size is the cardinality of its edge set. The vertices of
G corresponds to atoms, and an edge between two vertices is related to the chemical bond between these vertices. The degree of a vertex
u of a graph
G is symbolized by
, and is defined as the number of edges incident with
A graph is said to be regular, if all its vertices have the same degree, otherwise it is irregular. A sequence of non-negative integers
is called a degree sequence, if there exists a graph
G with
such as
. By
, we denote the number of vertices of degree
j for
The imbalance of an edge
is defined as
In [
17], Albertson initiated the term “irregularity of a graph
G” using the imbalance parameter as follows:
It is easy to see that a graph
G has irregularity zero if, and only if, it is a regular graph. Albertson [
17] showed that the irregularity of any graph is an even number. Furthermore, he proposed upper bounds on the irregularity of bipartite, triangle-free graphs, and for trees. The relations between irregularity and the matching number of trees and unicyclic graphs were examined in [
18]. Hansen et al. [
19] characterized the graphs with maximal irregularity. Abdo and Dimitrov [
20] worked out for the irregularity of graphs under several graph operations. This graph invariant is also known as the third Zagreb index. Recently, Abdo et al. [
21] introduced the new term “total irregularity measure of a graph G,” which was denoted and detailed as follows:
Dimitrov and
krekovski have introduced the relation between the irregularity and total irregularity of graphs in [
22]. The characterization of the graphs with extremal values of irregularity has been given by Tavakoli et al. in [
23]. The smallest graphs with equal irregularity measures are explored in [
24]. Fath-Tabar [
25] set up some new bounds on the Zagreb indices using the irregularity of graphs. Furthermore, Nasiri et al. [
26,
27], determined the second minimum of the irregularity and total irregularity indices in all graphs. Some new spectral bounds for graph irregularity have been given in [
28]. For the detail discussions about these graph invariants, we refer [
29,
30,
31,
32,
33,
34,
35]. Recently, Gutman et al. [
36] introduced the
irregularity index of a graph G, which is described as
number of elements in
G consisting of members and edge set
containing Some properties of this index have been presented in [
37,
38,
39]. If the vertex set and edge set of
G contains
and
elements, then the variance of
G is denoted by
and is defined as [
30]:
Although a lot of work has been executed on the degree and distance-based indices of chemical molecular graphs, the analyses of irregularity-related indices for chemical structures are still largely limited. In [
40,
41,
42], the irregularity indices of various chemical structures were examined. In this paper, we work out for the irregularity indices of the molecular graphs of distinct types of dendrimers.
2. Methodology
In this article, the schemes we used in our assessments are edge partition procedure, combinatorial registering, vertex partition technique, and degree-counting of vertices. Furthermore, we use Maple for calculations and Chemsketch for sketching the Figures.
Partition is an important topic in graph theory, and it also plays as a critical trick in many other graph problems in various settings. For example, labelling is a classical vertex partition problem which assigns all the vertices in graph a positive integer according to specific requirements, and it regard as a useful tool which widely applied in computer networks. Another example, harmonic coloring of graph is defined to color vertices with different colors satisfying: (1) two adjacent vertices have different colors; (2) the pair of colors for all edges in graph are different. That is to say, if , , , then . Therefore, the essence of harmonic coloring is an edge partition technique.
In our paper, we need to compute the irregularity measures of some specific dendrimers, and we need to divide the edge set into several classes in terms of the pair of degrees of two vertices of each edge. In each class, have the same value. This trick will be applied in computing , and . On the other hand, when it comes to , vertex partition approach is used in this article in which the whole vertex set is divided into several subsets according to their degrees in graph. Since the topological indices considered here are degree-based, the partition principles are degree-based designed as well.