# Molecular Irregularity Indices of Nanostar, Fullerene, and Polymer Dendrimers

Authors: Xie Qing; Zhen Wang; Mobeen Munir; Haseeb Ahmad
Journal: Journal of Chemistry
 (2020)

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## Abstract

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. The 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.


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## Body

## 1. 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]. These 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 [2–4]. Dendrimers are approximately monodisperse macromolecules consisting of symmetric branched units constructed around a small molecule or a linear polymer core [5–7]. Dendrimers are used in supermolecular chemistry, especially in hot-guest reactions and self-assembly processes. These are eminently defined artificial macromolecules, which are described by a combination of a high number of functional groups and a compact molecular structure [3]. These 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.Figure 1
Three main parts of a dendrimer.The formations of dendrimer molecules start with the central atom or group of atoms called core [5, 9–12]. There 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 [13–15].The 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 NS4[p], and polymer dendrimer NS5[p]. It has been keenly observed that the geometry and pattern of chemical systems characterize its physical aspects [16–24].


## 2. Preliminaries and Notations

LetG be a simple connected graph with vertex V and edge set E, and du 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. Their 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 Theory, Erdos officially posed an open problem to “ determine the extreme size of highly irregular graphs of given order” [26–28]. 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. The authors of [27] demonstrated that there does not exist any perfect graph. The 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. These irregularity indices have been studied recently in a novel way [31–33]. The first such irregularity index was introduced in [32]. Most of these indices used the concept of the imbalance of an edge defined as imballuv=du−dv [33]. The Albertson index, AL(G), was defined by Albertson in [32] as ALG=∑UV∈Edu−dv. In this index, the imbalance of edges is computed. The irregularity indices IRL(G) and IRLU(G) are introduced by Vukičević and Graovac [33] as IRLG=∑UV∈Elndu−lndv and IRLUG=∑UV∈Edu−dv/mindu,dv. Recently, Abdo et al. introduced the new term “total irregularity measure of a graph G”, which is defined as IRRtG=1/2∑UV∈Edu−dv [34–36]. Recently, Gutman introduced the IRF(G) irregularity index of the graph G, which is described as IRFG=∑UV∈Edu−dv2 in [37]. The Randic index itself is directly related to an irregularity measure, which is described as IRAG=∑UV∈Edu−1/2−dv−1/22 in [38]. Further irregularity indices of similar nature can be traced in [37, 39] in detail. These indices are given as IRDIFG=∑UV∈Edu/dv−dv/dv, IRLFG=∑UV∈Edu−dv/dudv, LAG=2∑UV∈Edu−dv/du+dv, IRD1=∑UV∈Eln1+dv−dv, IRGAG=∑UV∈Elndu+dv/2dudv, and IRBG=∑UV∈Edu1/2−dv1/22. Recently, Zahid et al. computed the irregularity indices of a nanotube [40]. Gao et al. recently computed irregularity measures of some dendrimer structures in [41] and molecular structures in [42]. Hussain et al. discussed irregularity indices of some well-known benzenoid systems in [43] and some classes of nanostar dendrimers NS1p, NS2p, and NS3p in [44]. Liu et al. computed Zagreb and multiple Zagreb indices of Eulerian graphs in [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 dendrimersDp, fullerene dendrimerNS4p, and polymer dendrimerNS5p. Figures 2–4 represent molecular graphs of these three systems. The 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]. The molecular pattern and topology of these three dendrimers are shown in these figures. Figure 2 represents the structure of Dp nanostar dendrimer. In Figures 3 and 4, structure of NS4p fullerene dendrimers and NS5p polymer dendrimer is shown, respectively.Figure 2
The nanostar dendrimerDp for p=1 shown in (a) and for p=2 shown in (b).
(a)
(b)Figure 3
Fullerene dendrimerNS4p.Figure 4
Polymer dendrimerNS5p.


## 3. Main Results

In this section, we present our main theoretical results.Theorem 1.
LetDp be the nanostar dendrimer, then the irregularity indices of Dp are as follows:(1)
IRDIFDp=55×2p−1−4.(2)
IRRDp=65×2p−1−4.(3)
IRLDp=2.4327906495×2p−1−4.(4)
IRLUDp=35×2p−1−4.(5)
IRLFDp=2.4494897435×2p−1−4.(6)
σDp=65×2p−1−4.(7)
IRLADp=2.45×2p−1−4.(8)
IRD1=4.1588830835×2p−1−4.(9)
IRADp=0.10102051445×2p−1−4.(10)
IRGADp=0.12246598365×2p−1−4.(11)
IRBDp=0.60612308665×2p−1−4.(12)
IRRtDp=35×2p−1−4.Proof.
In order to prove the above theorem, we have to consider Figure2 along with Table 1.
Now using Table1 and the above definitions, we have(1)IRDIFG=∑UV∈Edudv−dvdv,IRRDp,x,y=122×2p−1−122−22+65×2p−1−432−23+612×2p−1−933−33=65×2p−1−432−23,(2)IRRG=∑UV∈Eimbe∴imbe=du−dv,IRRDp,x,y=122×2p−1−12−2+65×2p−1−43−2+612×2p−1−93−3=65×2p−1−43−2,(3)IRLG=∑UV∈Eln du−ln dv,IRLDp,x,y=2×2p−1−1ln2−ln2+65×2p−1−4ln3−ln2+612×2p−1−9ln3−ln3=65×2p−1−4ln32,(4)IRLUG=∑UV∈Edu−dvmindu,dv,IRLUDp,x,y=2×2p−1−12−22+65×2p−1−43−22+612×2p−1−93−33=65×2p−1−43−22,(5)IRLFG=∑UV∈Edu−dvdudv,IRLFDp,x,y=2×2p−1−12−24+65×2p−1−43−26+612×2p−1−93−39=65×2p−1−43−26,(6)σG=∑UV∈Edu−dv2,σDp,x,y=2×2p−1−12−22+65×2p−1−43−22+612×2p−1−93−32ε2=65×2p−1−43−22,(7)IRLAG=2∑UV∈Edu−dvdu+dv,IRLADp,x,y=22×2p−1−12−24+65×2p−1−43−25+612×2p−1−93−36=265×2p−1−43−25,(8)IRD1=∑UV∈Eln1+dv−dv,IRD1Dp,x,y=2×2p−1−1ln1+2−2+65×2p−1−4ln1+3−2+612×2p−1−9ln1+3−3=65×2p−1−4ln2,(9)IRAG=∑UV∈Edu−1/2−dv−1/22,IRADp,x,y=2×2p−1−112−122+65×2p−1−413−122+612×2p−1−913−132=65×2p−1−413−122,(10)IRGAG=∑UV∈Elndu+dv2dudv,IRGADp,x,y=2×2p−1−1ln2+224+65×2p−1−4ln3+226+612×2p−1−9ln3+329=65×2p−1−4ln3+226,(11)IRBG=∑UV∈Edu1/2−dv1/22,IRBDp,x,y=2×2p−1−12−22+65×2p−1−43−22+612×2p−1−93−32=65×2p−1−43−22,(12)IRRtG=12∑u,v∈VGdu−dv,IRRtDp,x,y=12122×2p−1−12−2+65×2p−1−43−2+612×2p−1−93−3=35×2p−1−4.
Table2 shows the values of these irregularity indices for some test values of parameter p.Table 1
Edge partition of nanostar dendrimerDp.
Number of edgesdu,dv
Number of indices
(2,2)
12
2
×
2
p
−
1
−
1
(2,3)
6
5
×
2
p
−
1
−
4
(3,3)
6
12
×
2
p
−
1
−
9Table 2
Irregularity indices for nanostar dendrimerDp.
Irregularity indices
p
=
1
p
=
2
p
=
3
p
=
4
p
=
5
IRDIF
G
=
∑
U
V
∈
E
d
u
/
d
v
−
d
v
/
d
v
5
30
80
180
380
AL
G
=
∑
U
V
∈
E
d
u
−
d
v
6
36
96
216
456
IRL
G
=
∑
U
V
∈
E
ln d
u
−
ln d
v
2.4328
14.5968
38.9246
87.5805
184.8921
IRLU
G
=
∑
U
V
∈
E
d
u
−
d
v
/
min
d
u
,
d
v
3
18
48
108
228
IRLU
G
=
∑
U
V
∈
E
d
u
−
d
v
/
d
u
d
v
2.4495
14.6969
39.1918
88.1816
186.1612
IRF
G
=
∑
U
V
∈
E
d
u
−
d
v
2
6
36
96
216
456
IRLA
G
=
2
∑
U
V
∈
E
d
u
−
d
v
/
d
u
+
d
v
2.4
14.4
38.4
86.4
182.4
IRD
1
=
∑
U
V
∈
E
ln
1
+
d
v
−
d
v
4.1589
24.9533
66.5422
149.7198
316.0751
IRA
G
=
∑
U
V
∈
E
d
u
−
1
/
2
−
d
v
−
1
/
2
2
0.10102
0.6061
1.6163
3.6367
7.6775
IRGA
G
=
∑
U
V
∈
E
ln
d
u
+
d
v
/
2
d
u
d
v
0.1225
0.73479
1.9594
4.4087
9.3074
IRB
G
=
∑
U
V
∈
E
d
u
1
/
2
−
d
v
1
/
2
2
0.6061
3.6367
9.6979
21.8204
46.0653
IRR
t
G
=
1
/
2
∑
U
V
∈
E
d
u
−
d
v
3
18
48
108
228Theorem 2.
LetNS4p be the fullerene dendrimer, then the irregularity indices of NS4p are as follows:(1)
IRDIFNS4p=26.666.2p−1+2.66667.2p+1−0.6664.(2)
IRRNS4p=32.2p−1+2.2p+1−2.(3)
IRLNS4p=12.97488.2p−1+1.6986.2p+1−1.517628.(4)
IRLUNS4p=16.2p−1+2.2p+1−2.(5)
IRLFNS4p=13.063936.2p−1+1.1547.2p+1−1.533934.(6)
σNS4p=32.2p−1+4.2p+1−2.(7)
IRLANS4p=12.8.2p−1+2p+1−1.485714.(8)
IRD1=22.180704.2p−1+1.098612.2p+1−1.386294.(9)
IRANS4p=0.538784.2p−1+0.17863322p+1−0.128713.(10)
IRGANS4p=57.242453.2p−1+1.242453.2p−1+0.473936.(11)
IRBNS4p=3.232672.2p−1+0.535898.2p+1−0.377386.(12)
IRRtNS4p=16.2p−1+2p+1−1.Proof.
In order to prove the above theorem, we have to consider Figure3 and Table 3.
Now using Table3 and the above definitions, we have(13)IRDIFG=∑UV∈Edudv−dvdv,IRDIFNS4p,x,y=32.2p−1−832−23+2p+1+222−22+2p+131−13+86.33−33+643−34+344−44=32.2p−1−832−23+2p+131−13+643−34,(14)IRRG=∑UV∈Eimbe∴imbe=du−dv,IRRNS4p,x,y=32.2p−1−83−2+2p+1+22−2+2p+13−1+86.3−3+64−3+34−4=32.2p−1−8+2.2p+1+6,(15)IRLG=∑UV∈Eln du−ln dv,IRLNS4p,x,y=32.2p−1−8ln3−ln2+2p+1+2ln2−ln2+2p+1ln3−ln1+86.ln3−ln3+6ln4−ln3+3ln4−ln4=32.2p−1−8ln32+2p+1ln31+6.ln43,(16)IRLUG=∑UV∈Edu−dvmindudv,IRLUNS4p,x,y=32.2p−1−83−22+2p+1+22−22+2p+13−11+86.3−33+6.4−33+3.4−44=32.2p−1−82+2,2p+1+2,(17)IRLFG=∑UV∈Edu−dvdudv,IRLFNS4p,x,y=32.2p−1−83−26+2p+1+22−24+2p+13−13+86.3−39+6.4−312+3.4−416=32.2p−1−86+2.2p+13+612,(18)σG=∑UV∈Edu−dv2,σNS4p,x,y=32.2p−1−83−22+2p+1+22−22+2p+13−12+86.3−32+6.4−32+3.4−42=32.2p−1−8+4.2p+1+6,(19)IRLAG=2∑UV∈Edu−dvdu+dv,IRLANS3p,x,y=232.2p−1−83−25+2p+1+22−24+2p+13−14+86.3−36+6.4−37+3.4−48=232.2p−1−85+2p+12+67,(20)IRD1=∑UV∈Eln1+dv−dv,IRD1NS4n,x,y=32.2p−1−8ln1+3−2+2p+1+2ln1+2−2+2p+1ln1+3−1+86ln1+3−3+6ln1+4−3+3ln1+4−4=32.2p−1−8ln2+2p+1ln3+6ln2,(21)IRAG=∑UV∈Edu−1/2−dv−1/22,IRANS4p,x,y=32.2p−1−813−122+2p+1+212−122+2p+113−112+8613−132+614−132+314−142=32.2p−1−813−122+2p+113−112+614−132,(22)IRGAG=∑UV∈Elndu+dv2dudv,IRGANS4p,x,y=32.2p−1−8ln3+226+2p+1+2ln2+224+2p+1ln3+123+86ln3+329+6ln4+3212+3ln4+4216=32.2p−1−8ln526+2p+1ln423+6ln7212,(23)IRBG=∑UV∈Edu1/2−dv1/22,IRBNS4p,x,y=32.2p−1−83−22+2p+1+22−22+2p+13−12+863−32+64−32+34−42=32.2p−1−83−22+2p+13−12+64−32,(24)IRRtG=12∑UV∈Edu−dv,IRRtNS4p,x,y=1232.2p−1−83−2+2p+1+22−2+2p+13−1+863−3+64−3+34−4=1232.2p−1−8+2.2p+1+6.
Table4 demonstrates the values of these irregularity indices of fullerene dendrimers for some test values of parameter p.Table 3
Edge partition of fullerene dendrimers.
Number of edgesdu,dv
Number of indices
(2,3)
32.2
p
−
1
−
8
(2,2)
2
p
+
1
+
2
(1,3)
2
p
+
1
(3,3)
86
(3,4)
6
(4,4)
3Table 4
Irregularity indices for fullerene dendrimerNS4p.
Irregularity indices
p
=
1
p
=
2
p
=
3
p
=
4
p
=
5
IRDIF
G
=
∑
U
V
∈
E
d
u
/
d
v
−
d
v
/
d
v
36.66628
73.99896
148.66432
297.99504
596.65648
AL
G
=
∑
U
V
∈
E
d
u
−
d
v
38
78
158
318
638
IRL
G
=
∑
U
V
∈
E
lnd
u
−
lnd
v
18.251652
38.0209
77.5594
156.636
314.7908
IRLU
G
=
∑
U
V
∈
E
d
u
−
d
v
/
min
d
u
,
d
v
22
46
94
190
382
IRLU
G
=
∑
U
V
∈
E
d
u
−
d
v
/
d
u
d
v
16.148802
33.83153
69.197010
139.92795
281.38984
IRF
G
=
∑
U
V
∈
E
d
u
−
d
v
2
46
94
190
382
766
IRLA
G
=
2
∑
U
V
∈
E
d
u
−
d
v
/
d
u
+
d
v
15.31428
32.11428
65.714286
132.91428
267.31428
IRD
1
=
∑
U
V
∈
E
ln
1
+
d
v
−
d
v
25.18885
51.7640
104.9143
211.2149
423.8161
IRA
G
=
∑
U
V
∈
E
d
u
−
1
/
2
−
d
v
−
1
/
2
2
1.1246038
2.3779206
4.8845542
9.8978214
19.924355
IRGA
G
=
∑
U
V
∈
E
ln
d
u
+
d
v
/
2
d
u
d
v
58.958842
117.44374
234.4135
468.3531
936.2324
IRB
G
=
∑
U
V
∈
E
d
u
1
/
2
−
d
v
1
/
2
2
4.99887
10.375142
21.127670
42.6327
85.642838
IRR
t
G
=
1
/
2
∑
U
V
∈
E
d
u
−
d
v
19
39
79
159
319Theorem 3.
LetNS5p be the polymer dendrimer, then the irregularity indices of NS5p are as follows:(1)
IRDIFNS5p=4.999998.2p+3+8.2p+1+3.(2)
IRRNS5p=6.2p+3+6.2p+1.(3)
IRLNS5p=2.43279.2p+3+3.295836.2p+1+0.8630546.(4)
IRLUNS5p=3.2p+3+6.2p+1+3.(5)
IRLFNS5p=4.242641.2p+3+2.44949.2p+1−1.793151.(6)
σNS5p=6.2p+3+12.2p+1+6.(7)
IRLANS5p=2.4.2p+3+3.2p+1+0.6.(8)
IRD1=4.158882.2p+3+3.295836.2p+1−0.863046.(9)
IRANS5p=0.101022.2p+3+0.535899.2p+1+0.434877.(10)
IRGANS5p=10.87302.2p+3+3.727359.2p+1−7.145661.(11)
IRBNS5p=0.606126.2p+3+1.607694.2p+1+1.001568.(12)
IRRtNS5p=3.2p+3+3.2p+1.Proof.
In order to prove the above theorem, we have to consider Figure4 and Table 5.
Now, using Table5 and the above definitions, we have(25)IRDIFG=∑UV∈Edudv−dvdv,IRDIFNS5p,x,y=62p+3−132−23+62p+122−22+2433−33+32p+1+131−13=62p+3−132−23+32p+1+131−13,(26)IRRG=∑UV∈Eimbe∴imbe=du−dv,IRRNS5p,x,y=62p+3−13−2+62p+12−2+243−3+32p+1+13−1=62p+3−1+62p+1+1,(27)IRLG=∑UV∈Eln du−ln dv,IRLNS5p,x,y=62p+3−1ln3−ln2+62p+1ln2−ln2+24ln3−ln3+32p+1+1ln3−ln1=62p+3−1ln32+32p+1+1ln31,(28)IRLUG=∑UV∈Edu−dvmindudv,IRLUNS5p,x,y=62p+3−13−22+62p+12−22+24.3−33+32p+1+13−11=32p+3−1+62p+1+1,(29)IRLFG=∑UV∈Edu−dvdudv,IRLFNS5p,x,y=62p+3−13−26+62p+12−24+243−39+32p+1+13−13=62p+3−12+62p+1+16,(30)σG=∑UV∈Edu−dv2,σNS5p,x,y=62p+3−13−22+62p+12−22+24.3−32+32p+1+13−12=62p+1−1+122p+1+1,(31)IRLAG=2∑UV∈Edu−dvdu+dv,IRLANS5p,x,y=262p+3−13−23+2+62p+12−22+2+24.3−33+3+32p+1+13−13+1=262p+3−15+62p+1+14,(32)IRD1=∑UV∈Eln1+dv−dv,IRD1NS5n,x,y=62p+3−1ln1+3−2+62p+1ln1+2−2+24.ln1+3−3+32p+1+1ln1+3−1=62p+3−1ln2+32p+1+1ln3,(33)IRAG=∑UV∈Edu−1/2−dv−1/22,IRANS5p,x,y=62p+3−113−122+62p+112−122+24.13−132+32p+1+113−112=62p+3−113−122+32p+1+113−112,(34)IRGAG=∑UV∈Elndu+dv2dudv,IRGANS5p,x,y=62p+3−1ln3+226+62p+1ln2+224+24.ln3+329+32p+1+1ln3+123=62p+3−1ln526+32p+1+1ln423,(35)IRBG=∑UV∈Edu1/2−dv1/22,IRBNS5p,x,y=62p+3−13−22+62p+12−22+24.3−32+32p+1+13−12=62p+3−13−22+32p+1+13−12,(36)IRRtG=12∑UV∈Edu−dv,IRRtNS5p,x,y=1262p+3−13−2+62p+12−2+24.3−3+32p+1+13−1=1262p+3−1+62p+1+1.
Table6 depicts the values of these irregularity indices for some test values of parameter p.Table 5
Edge partition of polymer dendrimerNS5p.
Number of edgesdu,dv
Number of indices
(2,3)
6
2
p
+
3
−
1
(2,2)
6
2
p
+
1
(3,3)
24
(1,3)
3
2
p
+
1
+
1Table 6
Irregularity indices for polymer dendrimerNS5p.
Irregularity indices
p
=
1
p
=
2
p
=
3
p
=
4
p
=
5
IRDIF
G
=
∑
U
V
∈
E
d
u
/
d
v
−
d
v
/
d
v
114.9999
226.999
450.99
898.999
1794.99
AL
G
=
∑
U
V
∈
E
d
u
−
d
v
120
240
480
960
1920
IRL
G
=
∑
U
V
∈
E
lnd
u
−
lnd
v
52.97103
105.0790
209.2949
417.7269
834.5907
IRLU
G
=
∑
U
V
∈
E
d
u
−
d
v
/
min
d
u
,
d
v
75
147
291
579
1155
IRLU
G
=
∑
U
V
∈
E
d
u
−
d
v
/
d
u
d
v
75.8870
153.567
308.9277
619.648
1241.090
IRF
G
=
∑
U
V
∈
E
d
u
−
d
v
2
150
294
582
1158
2310
IRLA
G
=
2
∑
U
V
∈
E
d
u
−
d
v
/
d
u
+
d
v
51
101.4
202.2
403.8
807
IRD
1
=
∑
U
V
∈
E
ln
1
+
d
v
−
d
v
78.8624
158.5878
318.038
636.9406
1274.744
IRA
G
=
∑
U
V
∈
E
d
u
−
1
/
2
−
d
v
−
1
/
2
2
4.19482
7.95477
15.47466
30.514461
60.59404
IRGA
G
=
∑
U
V
∈
E
ln
d
u
+
d
v
/
2
d
u
d
v
181.73209
370.6098
748.3653
1503.876
3014.898
IRB
G
=
∑
U
V
∈
E
d
u
1
/
2
−
d
v
1
/
2
2
17.1303
33.2591
65.5167
130.0319
259.062
IRR
t
G
=
1
/
2
∑
U
V
∈
E
d
u
−
d
v
60
120
240
480
960


## 4. 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 sizep from 1 to 30.Figure5 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.Figure 5
Graphs of irregularity index IRDIF.In Figure6, curves of irregularity index IRR have been drawn against step size p and behavior of this index is similar to IRIDF.Figure 6
Curves of irregularity index IRR.All irregularity indices show the similar pattern for three dendrimers, as indicated in Figures7–9.Figure 7
Curves of irregularity indexσ.Figure 8
Curves of irregularity index IRGA.Figure 9
Curves of irregularity index IRRt.From the above discussion, it can be concluded that NS5[p] is relatively more irregular than NS4[p] and D[p] is most regular in these three dendrimers. The dependence of irregularity of structures on step size could be a potential information career for modelling nanoscale drugs and devices.



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