Startseite Naturwissenschaften Eccentricity based topological indices of siloxane and POPAM dendrimers
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Eccentricity based topological indices of siloxane and POPAM dendrimers

  • Muhammad Azhar Iqbal , Muhammad Imran EMAIL logo und Muhammad Asad Zaighum
Veröffentlicht/Copyright: 21. Juni 2020
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Main Group Metal Chemistry
Aus der Zeitschrift Main Group Metal Chemistry Band 43 Heft 1

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.

Keywords: siloxane dendrimers; POPAM dendrimers; eccentricity based indices

1 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 structure-property 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:Gl→ R is a function such that for any Gl,Gm∈G, Top(Gl) = Top(Gm) when Gl and Hm 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 dG(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:

ξ(G)=∑W∈V(G)d(w)⋅ε(w)

and also the total eccentricity index of G presented by:

ζ(G)=∑u∈V(G)ε(u)

For further aspects of these beneficial indices, please see: Ashrafi et al. (2009a), Ashrafi et al. (2009b), Ashrafi and Sadati (2009), Ashrafi and Saheli (2010), Akhter et al. (2018), Akhter and Imran (2016), Gao et al. (2019), Gupta et al. (2002), Iqbal et al. (2018), Iqbal et al. (2019), Yang et al. (2019), and Zhou and Du (2010).

Zagreb eccentricity indices are investigated by (Ghorbani and Hosseinzadeh, 2012), and are expressed as:

M1*(G)=∑w∈V(G)(ε(w))2M2*(G)=∑wx∈E(G)ε(w)⋅ε(x)

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 nano-structure has been a great interest of many researchers in chemical graph theory.

2 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).

ABC(SD[r])=(33+32)2r−3−32,GA(SD[r])=(245+42)2r−85−42,ABC(PD[r])=(16×2r−1122),GA(PD[r])=(832+16+2456)2r−5−1256

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 3×2r+2−7 and 3×2r+2−8, respectively.

Figure 1 SD1[r] for r = 1 and r = 3.
Figure 1

SD1[r] for r = 1 and r = 3.

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 2r+5−10 and 2r+5−11, respectively.

Figure 2 PD[r] with r = 1 and r = 3.
Figure 2

PD[r] with r = 1 and r = 3.

3 Main discussion and results

We start this section by computing the eccentric-connectivity index of SD1[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:

ξ(SD[r])=(96r-18)2r−32r+30

Proof

As the structure of SD1[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, um, vm, wm for 1 ≤ m ≤ r, and are presented in Table 1, along their rate of repetition.

Table 1

The characterizations of elements of V(SD[r]).

RepresentativesDegreeEccentricityRate of repetition
U12r + 21
V42r + 11
W22r + 23
um12r + 2m + 23 × 2m−1
vm42r + 2m + 13 × 2m−1
wm2(1 ≤ m ≤ r−1)2r + 2m + 23 × 2m
wr14r + 23 × 2r

Using Table 1, the eccentric-connectivity index of SD[r] for r ≥ 1, can be picked up as follows:

ξ(SD[r])=∑u∈  V(SD1[  ])(u)⋅ε(u)              =1×1×(2r+2)+4×1×(2r+1)+2×3×(2r+2)              +∑=1r−1(1×3×2m−1(2r+2m+2)+4×3×2m−1(2r+2m+1)+2×3×2m(2r+2m+2))              +1×3×2r−1(4r+2)+4×3×2r−1(4r+1)+1×3×2r(4r+2)

After simplification, we get:

ξ(SD[r])=(96r−18)2r−32r+30

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:

ξ(PD[r])=(256r−32)2r−88r+126

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, am, bm, cm, dm for 1 ≤ m ≤ r are shown in Table 2 with their rate of repetition.

Table 2

The characterizations of elements of V(PD[r]).

RepresentativesVertex degreeEccentricityFrequency
x24r + 32
y24r + 42
z34r + 52
ai24r + 4m + 22m+1
bi24r + 4m + 32m+1
ci24r + 4m + 42m+1
di3(1 ≤ m ≤ r−1)4r + 4m + 52m+1
dr18r + 52r+1

Using the statistics presented in Table 2, the eccentric-connectivity index of PD[r], for r ≥ 1, can be evaluated as:

ξ(PD[r])=2×2(4r+3)+2×2(4r+4)+3×2×(4r+5)               +∑m=1r−1[2×2m+1×(4r+4m+2)+2×2m+1×(4r+4m+3)+2×2m+1×(4r+4m+4)+3×2m+1×(4r+4m+5)]               +2×2r+1×(8r+2)+2×2r+1×(8r+3)+2×2r+1(8r+4)                      +1×2r+1(8r+5)ξ(SD[r])=(96r−18)2r−32r+30

which completes the proof.

Comparison of the eccentric-connectivity index ξ(SD[r]) and PD[r] is shown in Figure 3. It is easy to see that in the given domain the eccentric connectivity index of PD[r] is more dominating.

Figure 3 Comparison of the eccentric-connectivity index ξ(SD[r]) and PD[r]. The colors red and blue represents the eccentric-connectivity index ξ(SD[r]) and PD[r], respectively. We can see that in the given domain the eccentric connectivity index of PD[r] is more dominating.
Figure 3

Comparison of the eccentric-connectivity index ξ(SD[r]) and PD[r]. The colors red and blue represents the eccentric-connectivity index ξ(SD[r]) and PD[r], respectively. We can see that in the given domain the eccentric connectivity index of PD[r] is more dominating.

The upcoming two results can be easily established from Tables 1 and 2.

Corollary 3.1

The total-eccentricity index of siloxane nanostar dendrimers SD[r] is:

ζ(SD[r])=(48r−3)2r−14r+12

Corollary 3.2

The total-eccentricity index of PD[r] nanostar dendrimers is:

ζ(PD[r])=(64r−4)2r+1−40r+32

Figure 4 gives a comparison of the total-eccentricity index of ς(SD[r]) and PD[r]. It is clear from Figure 4 that in the given domain the total-eccentricity index of ξ(SD[r]) is more dominating.

Figure 4 Comparison of the total-eccentricity index of ς(SD[r]) and PD[r]. The colors green and blue represents the total-eccentricity index of ς(SD[r]) and PD[r], respectively. We can see that in the given domain the total-eccentricity index of ξ(SD[r]) is more dominating.
Figure 4

Comparison of the total-eccentricity index of ς(SD[r]) and PD[r]. The colors green and blue represents the total-eccentricity index of ς(SD[r]) and PD[r], respectively. We can see that in the given domain the total-eccentricity index of ξ(SD[r]) is more dominating.

Now, we compute Zagreb eccentricity indices of the SD[r] and PD[r] nanostar dendrimers.

Theorem 3.3

The first Zagreb-eccentricity index of siloxane nanostar dendrimer SD[r] is:

M1*(SD[r])=(192r2−24r+99)2r−28r2+48r−82

Proof

From Table 1, we apply the values and evaluate the first Zagreb-eccentricity index of SD[r] as follows:

M1⋆(SD[r])=∑v∈V(SD1 [  ])[ε(v)]2                  =1×(2r+2)2+1×(2r+1)2+3×(2r+2)2                     +∑m=1r(3×2m−1(2r+2m+2)2+3×2m−1(2r+2m+1)2+3×2m(2r+2m+2)2)                  =(192r2−24r+99)2r−28r2+48r-82

Theorem 3.4

The second Zagreb-eccentricity index of siloxane nanostar dendrimer SD[r] is:

M2∗SDr=168r2−54r+872r−20r2+66r−70

Proof

From Table 3, we apply the values and evaluate the second Zagreb-eccentricity index of SD[r] as follows:

Table 3

The edge distribution of SD[r] with reference to edges and their rate of occurrence. The eccentricities are picked up from Table 1.

EdgesEccentricitiesRate of repetition
[u,v][2r+2,2r+1]1
[v,w][2r+1,2r+2]3
[w,v1][2r+2,2r+3]3
[vm,um] (1 ≤ m ≤ r)[2r+2m+1,2r+2m+2]3×2m−1
[vm,wm] (1 ≤ m ≤ r)[2r+2m+1,2r+2m+2]3×2m
[wm,vm+1] (1 ≤ m ≤ r−1)[2r+2m+2,2r+2m+3]3×2m

M2⋆(SD[r])=∑uv∈    (SD1[  ])ε(u)⋅ε(v)                  =1×(2r+2)(2r+1)+3×(2r+2)                     +3×(2r+2)(2r+3)                 +∑m=1r(3×2m−1(2r+2m+1)(2r+2m+2)+3×2m(2r+2m+1)(2r+2m+2))                 +∑m=1r−1(3×2m(2r+2m+2)(2r+2m+3))                 =(168r2−54r+87)2r−20r2+66r−70

This completes the result.

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 M2* (SD[r]) index of siloxane nanostar dendrimer SD[r] is more dominating.

Figure 5 Comparison of first and second Zagreb-eccentricity index of siloxane nanostar dendrimer SD[r]. The colors blue and green represents the first and second Zagreb-eccentricity index of siloxane nanostar dendrimer SD[r], respectively. We can see that in the given domain the MSD[r]) index of siloxane nanostar dendrimer SD[r] is more dominating.
Figure 5

Comparison of first and second Zagreb-eccentricity index of siloxane nanostar dendrimer SD[r]. The colors blue and green represents the first and second Zagreb-eccentricity index of siloxane nanostar dendrimer SD[r], respectively. We can see that in the given domain the MSD[r]) index of siloxane nanostar dendrimer SD[r] is more dominating.

Theorem 3.5

The first Zagreb-eccentricity index of PD2 [r] is:

M1* (PD[r]) = (512r2 − 64r + 268)2r+1 − 160r2 + 448r − 148

Proof

We apply the values of Table 2 and evaluate the first Zagreb-eccentricity index of PD[r] as:

M1⋆PDr=2×4r+32+2×4r+42+2×4r+52+∑m=1r2m+1×4r+4m+22+2m+1×4r+4m+32+2m+1×4r+4m+42+2m+1×4r+4m+52=512r2−64r+2682r+1−160r2+256r−436

which completes the result.

Theorem 3.6

The second Zagreb-eccentricity index of PD[r] is:

M2*(PD[r])=(192r2−24r+98)22+r+(256r2−160r+152)2r−176r2                             +280r−471

Proof

We compute the second Zagreb-eccentricity index of PD[r] by use of Table 4 as:

Table 4

The edge distribution of PD[r] with reference to edges and their rate of occurrence. The eccentricities are picked up from Table 2.

EdgesEccentricitiesRate of repetition
[x,x][4r + 3,4r + 3]1
[x,y][4r + 3,4r + 4]2
[y,z][4r + 4,4r + 5]2
[z,a1][4r + 5,4r + 6]4
[am,bm] (1 ≤ m ≤ r)[4r + 4m + 2,4r + 4m + 3]2m+1
[bm,cm] (1 ≤ m ≤ r)[4r + 4m + 3,4r + 4m + 4]2m+1
[cm,dm] (1 ≤ m ≤ r)[4r + 4m + 4,4r + 4m + 5]2m+1
[dm,am+1] (1 ≤ m ≤ r−1)[4r + 4m + 5,4r + 4m + 6]2m+2

M2⋆(PD[r])=1×(4r+3)(4r+3)+2×(4r+3)(4r+4)                     +2×(4r+4)(4r+5)+4×(4r+5)(4r+6)                     +∑m=1r[2m+1(4r+4m+2)(4r+4m+3)+2m+1(4r+4+3)(4r+4+4)+2m+1(4r+4m+4)(4r+4m+5)]                     +∑m=1r−12m+2(4r+4m+5)(4r+4m+6)                      =(192r2−24r+98)22+r+(256r2−160r+152)2r                      −176r2+280r−471

This gives the required result.

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 M1* (PD[r]) is more dominating.

Figure 6 Comparison of first Zagreb-eccentricity index of PD[r] and second Zagreb eccentricity index of PD[r]. The colors red and blue represents the first and second Zagreb-eccentricity indices of PD[r], respectively. We can see that in the given domain M*1(PD[r]) is more dominating.
Figure 6

Comparison of first Zagreb-eccentricity index of PD[r] and second Zagreb eccentricity index of PD[r]. The colors red and blue represents the first and second Zagreb-eccentricity indices of PD[r], respectively. We can see that in the given domain M*1(PD[r]) is more dominating.

All above results are concluded in the following Table 5.

Table 5

Concluded results.

IndicesResults
ξ(SD[r])(96r − 18)2r − 32r + 30
ξ(PD[r])(256r − 32)2r − 88r + 126
ς(SD[r])(48r − 3)2r − 14r + 12
ς(PD[r])(64r − 4)2r+1 − 40r + 32
M(SD[r])(192r2 − 24r + 99)2r − 28r2 + 48r − 82
M(SD[r])(168r2 − 54r + 87)2r − 20r2 + 66r − 70
M(PD[r])(512r2 − 64r + 268)2r+1 − 160r2 + 448r − 148
M(PD[r])(192r2 − 24r + 98)22+r + (256r2 − 160r + 152)2r − 176r2 + 280r − 471

4 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.

Acknowledgements

This research is supported by the the UPAR Grant of United Arab Emirates University (UAEU), Al Ain, United Arab Emirates via Grant No. G00002590 and UPAR Grant of UAEU via Grant No. G00003271. The authors are very grateful to the referees for their constructive suggestions and useful comments, which improved this work very much.

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Received: 2020-02-08
Accepted: 2020-04-27
Published Online: 2020-06-21

© 2020 Iqbal et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  22. A new 2D dibutyltin coordination polymer with 3,5-dinitrosalicylate and 4,4’-bipyridine ligands
  23. Review
  24. Structures of Pt(0)P3, Pt(0)P4 and Pt(II)P4 – Distortion isomers
  25. Special Issue: Topological descriptors of chemical networks: Theoretical studies (Guest Editors: Muhammad Imran and Muhammad Javaid)
  26. Modified Zagreb connection indices of the T-sum graphs
  27. Topological properties of metal-organic frameworks
  28. Eccentricity based topological indices of siloxane and POPAM dendrimers
  29. On topological aspects of degree based entropy for two carbon nanosheets
  30. On multiplicative degree based topological indices for planar octahedron networks
  31. Computing entire Zagreb indices of some dendrimer structures
https://doi.org/10.1515/mgmc-2020-0010
Schlagwörter für diesen Artikel
siloxane dendrimers; POPAM dendrimers; eccentricity based indices
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Research Articles
  2. An accelerated and effective synthesis of zinc borate from zinc sulfate using sonochemistry
  3. A new approach on lithium-induced neurotoxicity using rat neuronal cortical culture: Involvement of oxidative stress and lysosomal/mitochondrial toxic Cross-Talk
  4. Green synthesis and characterization of hexaferrite strontium-perovskite strontium photocatalyst nanocomposites
  5. Assessment of content and chemical forms of arsenic, copper, lead, and chromium in sewage sludge compost as affected by various bulking agents
  6. Preparation of skeletally diverse quinazoline-2,4(1H,3H)-diones using Na2SiO3/SnFe2O4 catalytic system through a four-component reaction
  7. Efficient photocatalytic degradation of organic dye from aqueous solutions over zinc oxide incorporated nanocellulose under visible light irradiation
  8. Synthesis of pyrimidines by Fe3O4@SiO2-L-proline nanoparticles
  9. Abnormally aggregation-induced emissions observed from hydrogen- and silyl-substituted siloles
  10. Organodiphosphines in PtP2X2 (X = As, Ge or Te) derivatives – Structural aspects
  11. Synthesis and structural characterization of dialkyltin complexes of N-salicylidene-L-valine
  12. Ultrasound-promoted solvent-free synthesis of some new α-aminophosphonates as potential antioxidants
  13. Occupational exposure in lead and zinc mines induces oxidative stress in miners lymphocytes: Role of mitochondrial/lysosomal damage
  14. Eccentric topological properties of a graph associated to a finite dimensional vector space
  15. Magnetically recoverable nanostructured Pd complex of dendrimeric type ligand on the MCM-41: Preparation, characterization and catalytic activity in the Heck reaction
  16. Short Communications
  17. The crystal structure of the first ether solvate of hexaphenyldistannane [(Ph3Sn)2 • 2 THF]
  18. New crystal structures of alkali metal tetrakis(pentafluorophenyl)borates
  19. s-Block metal scorpionates – A new sodium hydrido-tris(3,5-dimethyl-1-pyrazolyl)borate salt showing an unusual core stabilized by bridging and terminal O-bonded DMSO ligands
  20. Reduction of a 1,4-diazabutadiene and 2,2’-bipyridine using magnesium(I) compounds
  21. fac-Bis(phenoxatellurine) tricarbonyl manganese(I) bromide
  22. A new 2D dibutyltin coordination polymer with 3,5-dinitrosalicylate and 4,4’-bipyridine ligands
  23. Review
  24. Structures of Pt(0)P3, Pt(0)P4 and Pt(II)P4 – Distortion isomers
  25. Special Issue: Topological descriptors of chemical networks: Theoretical studies (Guest Editors: Muhammad Imran and Muhammad Javaid)
  26. Modified Zagreb connection indices of the T-sum graphs
  27. Topological properties of metal-organic frameworks
  28. Eccentricity based topological indices of siloxane and POPAM dendrimers
  29. On topological aspects of degree based entropy for two carbon nanosheets
  30. On multiplicative degree based topological indices for planar octahedron networks
  31. Computing entire Zagreb indices of some dendrimer structures
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