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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access August 3, 2018

Calculating the Degree-based Topological Indices of Dendrimers

  • Shin Min Kang EMAIL logo , Manzoor Ahmad Zahid , Abaid ur Rehamn Virk , Waqas Nazeer and Wei Gao
From the journal Open Chemistry


Topological indices capture the symmetry of molecular structures and provide a mathematical language to predict properties such as boiling points, viscosity, the radius of gyrations etc. [1]. We compute some newly developed topological indices of Porphyrin, Propyl Ether Imine, Zinc-Porphyrin and Poly(EThyleneAmidoAmine) Dendrimers. The results are plotted using Maple 2015 software to see the dependence on the involved structural parameters.

1 Introduction

In chemical graph theory, a molecular graph is a simple graph having no loops and multiple edges in which atoms and chemical bonds are represented by vertices and edges respectively. A graph G(V, E) with vertex set V(G) and edge set E(G) is connected if there exists a connection between any pair of vertices in G. The degree of a vertex is the number of vertices which are connected to that fixed vertex by the edges. The distance between two vertices u and v is denoted as d(u, v) = dG(u, v) and is the length of the shortest path between u and v in graph G. The concept of degree is somewhat closely related to the concept of valence in chemistry. For details on the basics of graph theory any standard text such as [2] can be of great help.

Cheminformatics is another emerging field in which quantitative structure-activity (QSAR) and structure-property (QSPR) relationships predict the biological activities and properties of nanomaterials. In these studies, some physic-chemical properties and topological indices are used to predict bioactivity of the chemical compounds for example see [3, 4, 5, 6]. Algebraic polynomials have also useful applications in chemistry such as the Hosoya polynomial (also called Wiener polynomial) [7] which play a vital role in determining distance-based topological indices. Among other algebraic polynomials, M-polynomial was introduced by Deutsch and Klavzar [8], plays the same role in determining the closed form of many degree-based topological indices [9, 10, 11, 12, 13]. The main advantage of M-polynomial is the wealth of information that it contains about degree-based graph invariants.

Dendrimers are man-made, nanoscale compounds with unique properties that make them useful to the health and pharmaceutical industry as both enhancements to existing products and as entirely new products. Dendrimers are constructed by the successive addition of layers of branching groups. The final generation incorporates the surface molecules that give the dendrimer the desired function for pharmaceutical, life science, chemical, electronic and materials applications. Dendrimers fall under the broad heading of nanotechnology, which covers the manipulation of matter in the size range of 1-100 nanometers (one million nanometers equal one millimeter) to create compounds, structures, and devices with a novel, pre-determined properties.

Dendrimers are repetitively branched molecules. A dendrimer is typically symmetric around the core and often adopts a spherical three-dimensional morphology. The first dendrimers were made by divergent synthesis approaches by Buhleier et al. [14] in 1978. Dendrimers have gained a wide range of applications in chemistry, biology and nanosciences. There are a lot of research papers on the computation of topological indices of dendrimers, for example [9,15,16]. In this report, we discuss degree-based combinatorial facts about four familiar classes of dendrimers, Porphyrin, Propyl Ether Imine, Zinc-Porphyrin and Poly(EThyleneAmidoAmine) Dendrimers. It is important to remark that all dendrimers differ in cores.

2 Basic Definition and Literature Review

Weiner, in 1947, approximated the boiling point of alkanes as αW(G) + βP3 + γ where α, β and γ are empirical constants, W(G) is the Weiner index and P3 is the path of length 3 [17]. Thus Weiner laid the foundation of Topological index which is also known as connectivity index. A lot of chemical experiments require determining the chemical properties of emerging nanotubes and nanomaterials. Chemical-based experiments reveal that out of more than 140 topological indices no single index is strong enough to determine many physic-chemical properties, although, in combination, these topological indices can do this to some extent. The Wiener index is originally the first and most studied topological index, see for details [18,19].

Randić index, [20] denoted by R−1/2 (G) and introduced by Milan Randić in 1975 is also one of the oldest topological indexes. In 1998, working independently, Bollobas and Erdos [21] and Amic et al. [22] proposed the generalized Randić index and has been studied extensively by both chemists and mathematicians [23] and many mathematical properties of this index have been discussed in [24]. For a detailed survey we refer the book [25]. The general Randić index is defined as Rα(G)=uvE(G)1(dudv)α, and the inverse Randić index is defined as RRα(G)=uvE(G)(dudv)α. Obviously R−1/2(G) is the particular case of Rα(G) when α=12, and R−1/2(G) is the particular case of RRα (G) when α=12. The Randić index is the most studied among all other topological indices. Many papers and books such as [26,27,28] are written on this topological index. Randić himself wrote two reviews on his Randić index [29,30] and there are three more reviews [31,32,33]. The suitability of the Randić index for drug design was immediately recognized, and eventually, the index was used for this purpose on countless occasions. The physical reason for the success of such a simple graph invariant is still an enigma, although several more-or-less plausible explanations were offered.

Gutman and Trinajstić introduced first Zagreb index and second Zagreb index, which are defined as:




respectively. Futher detail about these indices can be found in references [34,35,36,37,38,39,40,41,42,43,44,45,46,47].

The sum connectivity index is defined as


and modified Randić index is defined as


Shigehalli and Kanabur [39] introduced following new degree-based topological indices: Arithmetic-Geometric (AG1) index AG1(G)=uvE(G)du+dv2dudv,

Shigehalli & Kanabur indices are defined as:


3 Main Results and Proofs

In this section we give our computational results.

Theorem 1

Let DnPn be the Porphyrin Dendrimer. Then

  1. χ(DnPn)=(6+725+136+87)n(52+65),

  2. R(DnPn)=34n92,

  3. AG1(DnPn)=(63+53+206)n5526,

  4. SK (DnPn) = 271n – 25,

  5. SK1(DnPn)=6432n28,

  6. SK2(DnPn)=713n1152.


For Porphyrin Dendrimer DnPn, we have |V (DnPn)| = 96n –10 and |E (DnPn)| = 105n –11.

There are six types of edges in E (DnPn) based on the degree of end vertices. i.e.,

E1 (DnPn) = {uvE(DnPn): du = 1, dv = 3};

E2 (DnPn) = {uvE(DnPn): du = 1,dv = 4};

E3 (DnPn) = {uvE(DnPn): du = 2, dv = 2};

E4 (DnPn) = {uvE(DnPn): du = 2, dv = 3};

E5 (DnPn) = {uvE(DnPn): du = 3, dv = 3};

E6 (DnPn) = {uvE(DnPn): du = 3, dv = 4}.

It can be observed from Figure 1 that |E1(DnPn)| = 2n, |E2 (DnPn)| = 24n, |E3(DnPn)| = 10n – 5, |E4 (DnPn)| = 48n – 6, |E5 (DnPn)| = 13n and |E6(DnPn)| = 8n.

Figure 1 Porphyrin Dendrimer.
Figure 1

Porphyrin Dendrimer.



Hence, the desired results are obtained.

Theorem 2

Let PETIM be the Propyl Ether Imine Dendrimer. Then

  1. χ(PETIM)=132n+13+122n+49+15(6.2n6)5;

  2. R′ (PETIM) = 11⋅2n-11 ;

  3. AG1(PETIM)=322n+12+22n+436+56(62n6)6;

  4. SK (PETIM) = 3⋅2n + 2n+5 + 15⋅2n – 51 ;

  5. SK1 (PETIM) = 2n+1 + 2n+5 + 18⋅2n – 54 ;

  6. SK2 (PETIM) = 9⋅2n−1 + 2n+6 + 75⋅2n−12192.


For Poly Ether Imine Dendrimer PETIM, we infer |V(PETIM)| = 24.2n – 23 and |E(PETIM)| = 24.2n – 24.

There are three types of edges in E (PETIM) based on the degree of end vertices. i.e.,

E1 (PETIM) = {uvE(PETIM) : du = 1, dv = 2};

E2 (PETIM) = {uvE(PETIM) : du = 2,dv = 2};

E3 (PETIM) = {uvE(PETIM) : du = 2, dv = 3}.

It can be observed from Figure 2 that |E1 (PETIM)| = 2n+1, |E2 (PETIM)| = 2n+4 –18 and |E3 (PETIM)| = 6.2n = 6.

Figure 2 Propyl Ether Imine Dendrimer.
Figure 2

Propyl Ether Imine Dendrimer.



Therefore, we finish the proof.

Theorem 3

Let DPZn be the Zinc-Porphyrin Dendrimer. Then

  1. χ(DPZn)=82n2+15(402n16)5+16(82n16)6+477;

  2. R(DPZn)=242n353.;

  3. AG1(DPZn)=242n20+512(402n16)6+733,;

  4. SK(DPZ)==156⋅2 -82 ;

  5. SK1 (DPZn)=188⋅2n -104;

  6. SK2 (DPZn)=386⋅2n -211.


For Zinc-Porphyrin Dendrimer DPZn, we deduce |V(DPZn)| = 96n–10 and |E(DnPn)| = 105n–11.

There are four types of edges in E (DPZn) based on the degree of end vertices. i.e.,

E1 (DPZn) = {uvE(DPZn): du = 2, dv = 2};

E2 (DPZn) = {uvE(DPZn): du = 2, dv = 3};

E3 (DPZn) = {uvE(DPZn): du = 3, dv = 3};

E4 (DPZn) = {uvE(DPZn): du = 3, dv = 4}.

It can be observed from Figure 3 that |E1 (DPZn)| = 16⋅2n −4, |E2 (DPZn)| = 40⋅2n −16, |E3 (DPZn)| = 8⋅2n −16 and |E4 (DPZn)| 4.

Figure 3 Zinc-Porphyrin Dendrimer.
Figure 3

Zinc-Porphyrin Dendrimer.

Therefore, in terms of the index definitions, we get


Theorem 4

Let PETAA be the Poly(EThyleneAmidoAmine) Dendrimer. Then

  1. χ(PETAA)=432n+102n5+15(202n9);

  2. R(PETAA)=4632n193;

  3. AG1(PETAA)32n2+23(42n2)+162n8+512(202n9)6;

  4. SK(PETAA)=962n852.;

  5. SK1 (PETAA) = 102⋅2n -46 ;

  6. SK2(PETAA)=2142n3854.


For PETAA Dendrimer, we derive that |V(PETAA)| = 44⋅2n −18 and |E(PETAA)| = 44⋅2n −19.

There are four type of edges in E(PETAA) based on the degree of end vertices. i.e.,

E1 (PETAA) = {uvE(PETAA): du = 1, dv = 2};

E2 (PETAA) = {uvE(PETAA): du = 1, dv = 3};

E3 (PETAA) = {uvE(PETAA): du = 2, dv = 2};

E4 (PETAA) = {uvE(PETAA): du = 2, dv = 3}.

It can be observed from Figure 4 that |E1 (PETAA) = 4⋅2n, |E2(PETAA) = 4⋅2n −2, |E3(PETAA)| = 16⋅2n −8 and |E4 (PETAA)| = 20⋅2n −9.

Figure 4 Poly(EThyleneAmidoAmine Dendrimer.
Figure 4

Poly(EThyleneAmidoAmine Dendrimer.

In light of the definition of indices, we have


In all, we get the conclusions.

4 Graphical Comparison and Conclusions

Graph theory as a tool has been widely applied to molecular topology theory of chemical computation. The results are of guiding significance to the engineering application. In this paper, several important chemical structures are considered, and by means of edge set dividing trick and molecular graph structure analysis, the accurate formulas of some important degree-based indices are calculated. Our calculated results, for example the Randić index is useful for determining physio–chemical properties of alkanes as noticed by chemist Melan Randić in 1975. He noticed the correlation between the Randić index R and several physico-chemical properties of alkanes, for examples enthalpies of formation, boiling points, chromatographic retention times, vapor pressure and surface areas. An example is shown in Figure 5, adapted from [20], relating the boiling point of some alkanes and their correlation with Randić index.

Figure 5 Correlation of Randić index with the boiling point of a selected set of alkanes.
Figure 5

Correlation of Randić index with the boiling point of a selected set of alkanes.

Now we give geometric comparison of the results. In figures 4-9, red, blue, lime and aqua colors are for Porphyrin, Propyl Ether Imine, Zinc-Porphyrin and Poly(EThyleneAmidoAmine) dendrimers respectively. Figures 6, 7 and 9 illustrate that the Poly(EThyleneAmidoAmine) dendrimer has greatest value of sum connectivity, Modify Randić and SK indices while the Porphyrin dendrimers have the lowest value of sum connectivity, Modify Randić and SK indices. From figure 8 and 11 we can observe that the Porphyrin dendrimers have the largest value of AGl and SK2 indices while Propyl Ether Imine dendrimers have the lowest s least value of AG1 and SK2 indices. From Figure 10 one can make observation that the SK1 index has largest value for Poly(EThyleneAmidoAmine) Dendrimers and least value for Propyl Ether Imine dendrimers.

Figure 6 Sum Connectivity Index.
Figure 6

Sum Connectivity Index.

Figure 7 Modify Randic Index.
Figure 7

Modify Randic Index.

Figure 8 AG Index.
Figure 8

AG Index.

Figure 9 SK Index.
Figure 9

SK Index.

Figure 10 SK1 Index.
Figure 10

SK1 Index.

Figure 11 SK2 Index.
Figure 11

SK2 Index.

  1. Author Contributions: All authors contribute equally in this article.

  2. Conflict of interest: The authors declare no conflict of interest.


[1] Rucker G., Rucker C., On topological indices, boiling points, and cycloalkanes, J. Chem. Inf. Comput. Sci., 1999, 39, 788–802.10.1021/ci9900175Search in Google Scholar

[2] West D. B., An Introduction to Graph Theory., Prentice-Hall, 2996.Search in Google Scholar

[3] Klavžar S., Gutman I., A Comparison of the Schultz molecular topological index with the Wiener index., J. Chem. Inf. Comput. Sci., 1996, 36, 1001–1003.10.1021/ci9603689Search in Google Scholar

[4] Brückler F. M., Došli´c T., Graovac A., Gutman I., On a class of distance-based molecular structure descriptors., Chem. Phys. Lett., 2011, 503, 336–338.10.1016/j.cplett.2011.01.033Search in Google Scholar

[5] Gao W., Wang Y.Q., Wang W.F., Shi, L., The first multiplication atom-bond connectivity index of molecular structures in drugs., Saudi Pharmaceutical Journal., 2017, 25(4), 548-555.10.1016/j.jsps.2017.04.021Search in Google Scholar PubMed PubMed Central

[6] Zhao B., Wu H.L., Pharmacological characteristics analysis of two molecular structures., Applied Mathematics and Nonlinear Sciences, 2017, 2(1), 93-110.10.21042/AMNS.2017.1.00008Search in Google Scholar

[7] Gao W., Wang Y.Q., Basavanagoud B., Jamil, M.K., Characteristics studies of molecular structures in drugs., Saudi Pharmaceutical Journal, 2017, 25(4), 580-586.10.1016/j.jsps.2017.04.027Search in Google Scholar PubMed PubMed Central

[8] Deutsch E., Klavzar S., M-Polynomial, and degree-based topological indices., Iran. J. Math. Chem., 2015, 6, 93–102.Search in Google Scholar

[9] Munir M., Nazeer W., Rafique S., Kang S. M., M-polynomial and related topological indices of Nanostar dendrimers., Symmetry, 2016, 8(9), 97, 10.3390/sym8090097.10.3390/sym8090097Search in Google Scholar

[10] Munir M., Nazeer W., Rafique S., Nizami A. R., Kang S. M., M-polynomial and degree-based topological indices of titania nanotubes., Symmetry, 2016, 8(11), 117, 10.3390/sym8110117.10.3390/sym8110117Search in Google Scholar

[11] Munir M., Nazeer W., Rafique S., Kang, S. M., M-polynomial and degree-based topological indices of polyhex nanotubes., Symmetry, 2016, 8(12), 149, 10.3390/sym8120149.10.3390/sym8120149Search in Google Scholar

[12] AjmalM., Nazeer W., Munir M., Kang S. M., Kwun Y. C., Some Algebraic Polynomials and Topological Indices of Generalized Prism and Toroidal Polyhex Networks, Symmetry, 2017, 9(1), 5, 10.3390/sym9010005.10.3390/sym9010005Search in Google Scholar

[13] Munir M., Nazeer W., Shahzadi S., Kang S. M., Some invariants of circulant graphs., Symmetry., 2016, 8(11), 134; 10.3390/sym8110134.10.3390/sym8110134Search in Google Scholar

[14] Buhleier E., Wehner W., Vgtle F., “Cascade”- and “Nonskid-Chainlike” Syntheses of Molecular Cavity Topologies., Synthesis, 1978, 2 (1978)155158, 10.1055/s-1978-24702.Search in Google Scholar

[15] Alikhani S., Hasni R., Arif H., On the atom-bond connectivity index of some families of dendrimers., J. Comput. Theor. Nanosci., 2014, 11, 1802-1805.10.1166/jctn.2014.3570Search in Google Scholar

[16] Ashrafi A. R., Mirzargar M., PI, Szeged and edge Szeged indices of an infinite family of nanostars dendrimers., Indian J. Chem., 2008, 47A, 538-541.Search in Google Scholar

[17] Wiener H., Structural determination of paraffin boiling points., J. Am. Chem. Soc, 1947, 69, 17–20.10.1021/ja01193a005Search in Google Scholar

[18] Dobrynin A. A., Entringer R., Gutman I., Wiener index of trees: theory and applications., Acta Appl. Math., 2001, 66, 211–249.10.1023/A:1010767517079Search in Google Scholar

[19] Gutman I., Polansky O. E., Mathematical Concepts in Organic Chemistry., Springer-Verlag New York, USA, 1986.10.1007/978-3-642-70982-1Search in Google Scholar

[20] Randic M., On the characterization of molecular branching., J. Am. Chem. Soc., 1975, 97, 6609- 6615.10.1021/ja00856a001Search in Google Scholar

[21] Bollobas B., Erdos P., Graphs of extremal weights., Ars. Combin., 1998, 50, 225–233.10.1016/S0012-365X(98)00320-3Search in Google Scholar

[22] Amic D., Beslo D., Lucic B., Nikolic S., Trinajstić N., The vertexconnectivity index revisited., J. Chem. Inf. Comput. Sci, 1998, 38, 819–822.10.1021/ci980039bSearch in Google Scholar

[23] Hu Y., Li X., Shi Y., Xu T., Gutman I., On molecular graphs with smallest and greatest zeroth-Corder general randic index., MATCH Commun. Math. Comput. Chem., 2005, 54, 425-434.Search in Google Scholar

[24] Caporossi G., Gutman I., Hansen P., Pavlovic L., Graphs with maximum connectivity index., Comput. Biol. Chem., 2003, 27, 85-90.10.1016/S0097-8485(02)00016-5Search in Google Scholar

[25] Li X., Gutman I., Mathematical Chemistry Monographs, No 1., Kragujevac, 2006.Search in Google Scholar

[26] Gao W., Wang W. F., The eccentric connectivity polynomial of two classes of nanotubes., Chaos, Solitons and Fractals, 2016, 89, 290-294.10.1016/j.chaos.2015.11.035Search in Google Scholar

[27] Gao W., Wang W. F., The fifth geometric arithmetic index of bridge graph and carbon nanocones., Journal of Difference Equations and Applications, 2017, 23(1-2), 100-109.10.1080/10236198.2016.1197214Search in Google Scholar

[28] Li X., Gutman I., Mathematical aspects of Randić-type molecular structure descriptors., Univ. Kragujevac, Kragujevac, 2006.Search in Google Scholar

[29] Randić M., On history of the Randić index and emerging hostility toward chemical graph theory., MATCH Commun. Math. Comput. Chem., 2008, 59, 5-124.Search in Google Scholar

[30] Randić M., The connectivity index 25 years after., J. Mol. Graphics Modell., 2001, 20, 19-35.10.1016/S1093-3263(01)00098-5Search in Google Scholar

[31] Gutman I., Furtula B., Recent Results in the Theory of Randić Index., Univ. Kragujevac, Kragujevac, 2008.Search in Google Scholar

[32] Li X., Shi Y., A survey on the Randić index., MATCH Commun. Math. Comput. Chem., 2008, 59, 127–156.Search in Google Scholar

[33] Li X., Shi Y., Wang L., in: Recent Results in the Theory of Randić Index, I. Gutman and B. Furtula (Eds.) 9-47, Univ. Kragujevac, Kragujevac, 2008.Search in Google Scholar

[34] Gao W., Yan L., Shi L., Generalized zagreb index of polyomino chains and nanotubes., Optoelectronics and Advanced Materials--Rapid Communications, 2017, 11(1-2), 119-124.Search in Google Scholar

[35] Basavanagoud B., Desai V. R., Patil S., (β, α)-Connectivity index of graphs., Applied Mathematics and Nonlinear Sciences, 2017, 2(1), 21-30.10.21042/AMNS.2017.1.00003Search in Google Scholar

[36] Sunilkumar M. H., Bhagyashri B. K., Ratnamma G. B., Vijay M. G., QSPR analysis of certain graph theocratical matrices and their corresponding energy., Applied Mathematics and Nonlinear Sciences, 2017, 2(1), 131-150.10.21042/AMNS.2017.1.00011Search in Google Scholar

[37] Trinajstic N., Nikolic S., Milicevic, A., Gutman I., On Zagreb indices., Kem. Ind., 2010, 59, 577-589.Search in Google Scholar

[38] Gao W., Siddiqui M. K., Molecular descriptors of nanotube, oxide, silicate, and triangulene networks., Journal of Chemistry, 2017, (2017), 6540754, in Google Scholar

[39] Shigehalli V. S., Kanabur R., Computation of new degree-based topological indices of graphene., Journal of Nanomaterials, 2016, (2016), 4341919.10.1155/2016/4341919Search in Google Scholar

[40] Sardar M. S., Pan X. F., Gao W., Farahani M. R., Computing Sanskruti Index of Titania Nanotubes., Open J. Math. Sci., 2017, 1(1), 126-131, in Google Scholar

[41] De N., Hyper Zagreb Index of Bridge and Chain Graphs., Open J. Math. Sci., 2018, 2(1), 01-17, in Google Scholar

[42] Riaz M., Gao M., Baig A. Q., M-Polynomials and degreebased Topological Indices of Some Families of Convex Polytopes., Open J. Math. Sci,, 2018, 2(1), 18-28, in Google Scholar

[43] Gao W., Muzaffar B., Nazeer W., K-Banhatti and K-Hyper Banhatti Indices of Dominating David Derived Network., Open J. Math. Anal., 2017, 1(1), 13-24.10.30538/psrp-oma2017.0002Search in Google Scholar

[44] Siddiqui H., Farahani M. R., Forgotten Polynomial and Forgotten Index of Certain Interconnection Networks., Open J. Math. Anal., 2017, 1(1), 45-60.10.30538/psrp-oma2017.0005Search in Google Scholar

[45] Tang Z., Liang L., Gao W., Wiener polarity index of quasi-tree molecular structures., Open J. Math. Sci., 2018, 2(1), 73-83, in Google Scholar

[46] Sardar M. S., Zafar S., Farahani M. R., The Generalized Zagreb Index of Capra-Designed Planar Benzenoid Series Cak(C6)., Open J. Math. Sci., 2017, 1(1), 44-51, in Google Scholar

[47] Mutee ur Rehman H., Sardar R., Raza A., Computing topological indices of Hex Board and its line graph., Open J. Math. Sci., 2017, 1(1), 62-71, in Google Scholar

Received: 2017-09-05
Accepted: 2018-05-03
Published Online: 2018-08-03

© 2018 Shin Min Kang et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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