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BY 4.0 license Open Access Published by De Gruyter Open Access September 25, 2019

Valency-based molecular descriptors of Bakelite network BNmn

  • Maqsood Ahmad , Muhammad Javaid , Muhammad Saeed and Chahn Yong Jung EMAIL logo
From the journal Open Chemistry


Bakelite network BNmnis a molecular graph of bakelite, a pioneering and revolutionary synthetic polymer (Thermosetting Plastic) and regarded as the material of a thousand uses. In this paper, we aim to compute various degree-based topological indices of a molecular graph of bakelite network BNmn. These molecular descriptors play a fundamental role in QSPR/QSAR studies in describing the chemical and physical properties of Bakelite network BNmn. We computed atom-bond connectivity ABC its fourth version ABC4 geometric arithmetic GA its fifth version GA5 Narumi-Katayama, sum-connectivity and Sanskruti indices, first, second, modified and augmented Zagreb indices, inverse and general Randic’ indices, symmetric division, harmonic and inverse sum indices of BNmn.

1 Introduction

In Chemical graph theory, chemical compounds are represented by graphs and mathematical techniques are used to solve problems arising in chemistry. A molecular graph is a simple connected graph in which atoms are taken as vertices and chemical bonds are taken as the edges of the graph. Topological indices are numerical numbers associated with molecular graphs of chemical

compounds and help us to predict the properties of chemical compounds without performing experiments and save money and time [1, 2]. Some applications related to topological indices of molecular graphs are given in [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

Throughout this paper, Γ is an ordered pair Γ=(V,E) where V is the set of vertices, and E is the set of edges. The order and size of a graph Γ is the cardinality of vertex set V and edge set E denoted by |V| and |E| respectively. Two vertices u and v are adjacent if e=uv is an edge in graph and e is said to be incident with u and v The valency of a vertex vΓ is the number of incident edges with v and is denoted by dv. The minimum and maximum valency of Γ is denoted by δ and Δ, respectively. The first topological index was introduced by Harold Wiener [18], when he was investigating the boiling point of alkane. Initially, this index was named as path number and now it is known as the Wiener index/number [19]. In 1975, after Wiener index, Milan Randic’ [20], introduced a prominent index known as Randić index and is defined as:


This simple graph invariant has found many applications in chemistry. Böllöbás and Erdös [21], extended the idea of Randić and proposed the general Randic’ index:


and inverse Randić index:


The additive version of Randic’ index is known as the sum-connectivity index (SCI) [22] and it gives a high correlation coefficient (0.99) for alkanes. The formula for SCI is given by:


Ivan Gutman and Trinajstić [23] proposed two topological indices named as the first and the second Zagreb indices which were denoted by M1 and M2 and got an immense attraction of researchers due to the applications in chemistry. First and second and Zagreb indices are defined as follows:


The modified second Zagreb index is defined as:


In 1998, Estrada et al. [24] established an important index named as the ABC(Γ) index, which is a good model to test stability of linear and branched alkanes and is defined below:


In 2009, Vukicevicet al. [25] introduced another remarkable index known as the geometric arithmetic index GA(Γ) and is given by the formula:


Three prominent and recently developed indices, denoted by ABC4(Γ), GA5(Γ) and S(Γ) are proposed by Ghorbani et al. [26], Graovac et al. [27], and Hosamani [28], respectively. These indices are different as compared to other vertex valency-based indices in the sense that they require partitioning of the edges of the network on the basis of neighbors’ valency-sum of end vertices for every edge in an interconnected network. Their formulae are given as:


where sv=wNΓ(v)dwand NΓ(v)={wV(Γ)|vwE(Γ)}.

Motivated by ABC index, Furtula et al. [29] offered the so called Augmented Zagreb index (AZI)

which proved to have a better correlation coefficient as compared to ABC [30] and is defined as:


A few more topological indices that have key importance are defined below which include harmonic index (HI), inverse sum index (ISI), and symmetric division index (SDD):


To compute topological indices, a huge mount of calculution is required. In 2015, Deutsch and Klavażr Klavażr [31] introduced M-polynomial to reduce this calculution


where mij(Γ) represent number of edges vwE(Γ) such that {dv(Γ), dw(Γ)}={i,j}. The other polynomials are Hosoya polynomial [32], matching polynomial [33], the Zhang-Zhang polynomial [34], the Schultz polynomial [35] and the Tutte polynomial [36]. Some promising topological indices are worked out in [31] with the help of M-polynomial and are depicted in the table 1 below.

Table 1

Formulae of certain essential topological descriptors in relation with M-polynomial.

Topological indicesFormulae derived from M-polynomial
1st Zagreb index (M1)(Dx+Dy)M(x,y)|x=y=1
2nd Zagreb index (M2)(Dx.Dy)M(x,y)|x=y=1
Modified 2nd Zagreb index (mM2)(Sx.Sy)M(x,y)|x=y=1
General Randic’ index Rα(Dxα.Dyα)M(x,y)|x=y=1
Inverse Randic’ index RRα(Sxα.Syα)M(x,y)|x=y=1
Symmetric Division Deg. Index (SDD)(DxSy+SxDy)M(x,y)|x=y=1
Harmonic Index (HI)2SxJM(x,y)|x=y=1
Inverse Sum Index (ISI)(SxJDxDy)M(x,y)|x=y=1
Augmented Zagreb Index (AZI)(Sx3Q2JDx3Dy3)M(x,y)
  1. where DxM=xMx,DyM=yMy,SxM=0xMt,ytdt,SyM=0yMx,ttdt,JMx,y=Mx,x,QαM=xαM.

In this paper, we computed all the above defined topological indices for Bakelite Network BNmn.

2 Bakelite Network BNmn

Synthetic polymers, such as Teflon, Bakelite, Polyvinyl chloride (PVC) and High-density polyethylene (HDPE) have been studied for a long time and are of industrial interest for diverse potential applications. For example, Bakelite

(C6H6OCH2O)n is very attractive and had immense influence on the development of technology. Since time immemorial natural materials like gum, amber, shellac, tortoise-shell, and horn have been used to make various type of tools and objects. Bakelite, invented by Belgian-American chemist Leo Hendrik Arthur Baekeland (1863-1944), was the very first synthetic material that brought mankind in to the plastic-age. Bakelite, being phenolic resin, possesses several desirable properties like water, heat, organic solvent and scratch resistant, hard, insoluble, rigid, and low-conductivity [37]. This revolutionary thermosetting plastic has enormous engineering applications and is used as an electric insulator, propellers, automotive accessories, and medical as well as sports equipment [38, 39].

3 Methodology

To compute our results, we count the number of vertices and number of edges of the graph of the Bakelite Network BNmn. The total number of vertices and edges in the graph of Bakelite network BNmnare 8mn-n+m and 10mn-2n , respectively. In the graph of BNmn, figure 3, n represent number of hexagons in one row and m represent number of hexagons in one column. The vertex set and edge set partition of any graph Γ can generally be defined as [3]

Figure 1 Structural phases of Bakelite.
Figure 1

Structural phases of Bakelite.

Figure 2 Molecular graph of Bakelite.
Figure 2

Molecular graph of Bakelite.

Figure 3 Hydrogen depleted molecular graph of Bakelite network B Nmn$B\ N_{m}^{n}$.
Figure 3

Hydrogen depleted molecular graph of Bakelite network BNmn.


We divide the edge set of graph of Bakelite Network BNmn.into classes depending on the degrees of end vertices of each edge. Then using this edge partition, we compute our desired results.

4 Computational Results

In this section, we compute various topological indices of bakelite network BNmn. In subsequent theorems, we provide the M-polynomial of a bakelite network which will be a core component of the forthcoming theorems.

Theorem 1

LetΓ=BNmnbe the (m,n)-dimensional bakelite network then M-polynomial of Γ is



Proof. By analyzing the molecular graph of bakelite network BNmnit can readily be observed that there are following three kind of vertices based on the degrees,


Such that


Now based on the degree of end vertices, the edge set of the molecular graph of bakelite network BNmncan be divided into following classes:


Such that





The next theorem is about the computation of nine indices from the M-polynomial.

Theorem 2

For bakelite networkΓ=BNmnclosed form formulae for first, second, modified and augmented Zagreb indices, Randic’indices, symmetric division degree, harmonic and inverse sum indices are:


Proof. Consider



  1. First Zagreb index

  2. Second Zagreb index

  3. Modified second Zagreb index

  4. Generalized Randić index

  5. Inverse Randić index

  6. Symmetric division index

  7. Harmonic index

  8. Inverse sum index

  9. Augmented Zagreb index


Theorem 3

LetΓ=BNmnbe the molecular graph of (m, n)-dimensional bakelite network, then




Theorem 4

LetΓ=BNmnbe the (m, n)-dimensional bakelite network then the fourth atom-bound connectivity index is given by


Proof. In order to prove our theorem, we need edge partition of Γ based on neighbors’ valency-sum of end vertices ∀vw∈Γ. We identify following nine categories of edges on valency based sum of neighbors’ vertices of each edge in the bakelite network.


such that


Now, the fourth version of ABC can be calculated as follows:


5 Conclusions

QSARs represent predictive models derived from the application of statistical tools correlating to the biological activity (including desirable therapeutic effect and undesirable side effects) of chemicals (drugs/toxicants/environmental pollutants) with descriptors representative of molecular structure and/or properties. QSARs are being applied in many disciplines like risk assessment, toxicity prediction, and regulatory decisions in addition to drug discovery and lead optimization. In this article, we obtained numerous molecular descriptors for a molecular graph of a pioneer synthetic polymer called bakelite. We employed edge partitioning procedures on the molecular graph on the basis of valency of vertices dv and sum of valency sv of vertices at unit distance from each other. In addition, we computed M-polynomial of bakelite network BNmnalong with closed form formulae of various valency-based topological descriptors of substantial significance. Moreover, we took advantage of Matlab and Maple for the simplification and plotting of results. For a future prospect, we propose to develop molecular graphs of certain synthetic polymers (plastics) like polyvinyl chloride (PVC) and Polyethylene terephthalate (PETE), and work out their topological descriptors which will eventually take part in determining a relation as well as comparison between the physical and chemical properties of these synthetic polymers with bakelite.


Authors are thankful to reviewers for valuable suggestions that improve the quality of this paper.

  1. Data Availability Statement: All data required for this research is available in this paper.

  2. Author Contribution: All authors contributed equally in this paper.

  3. Funding Statement: This research is partially funded by University of Management and Technology, Lahore, Pakistan.

  4. Completing Interest: The authors do not have any competing interests.


[1] Brückler F.M., Došlić 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

[2] Rücker G., Rücker C., On topological indices, boiling points, and cycloalkanes. J. Chem. Inf. Comput. Sci., 1999, 39, 788802.10.1021/ci9900175Search in Google Scholar

[3] Ramane H. S., Jummannaver R. B., Note on forgotten topological index of chemical structure in drugs, AMNS, 2016, 1(2), 369-374.10.21042/AMNS.2016.2.00032Search in Google Scholar

[4] De N., Hyper Zagreb Index of Bridge and Chain Grpahs, Open J. Math. Sci., 2018, 2(1), 1-17.10.30538/oms2018.0013Search in Google Scholar

[5] Tang Z., Liang L., Gao W., Wiener polarity index of quasi-tree molecular structures, Open J. Math. Sci., 2018, 2(1), 73-83.10.30538/oms2018.0018Search in Google Scholar

[6] Baig A.Q., Naeem M., Gao W., Revan and hyper-Revan indices of Octahedral and icosahedral networks, AMNS, 2018, 3(1), 33-40.10.21042/AMNS.2018.1.00004Search in Google Scholar

[7] Liu G., Jia Z., Gao W., Ontology similarity computing based on stochastic primal dual coordinate technique, Open J. Math. Sci., 2018, 2(1), 221-227.10.30538/oms2018.0030Search in Google Scholar

[8] Sardar M.S., Zafar S., Farahani M.R., The Generalized Zagreb Index of Capra-Designed Planar Benzenoid Series C a k ( C 6 ), Open J. Math. Sci., 2017, 1(1), 44-51.10.30538/oms2017.0005Search in Google Scholar

[9] Sardar M.S., Pan X.-F., Gao W., Farahani, M.R., Computing Sanskruti Index of Titania Nanotubes, Open J. Math. Sci., 2017, 1(1), 126-131.10.30538/oms2017.0012Search in Google Scholar

[10] Naeem M., Siddiqui M. K., Guirao J. L. G., Gao W., New and Modified Eccentric Indices of Octagonal Grid Omn, AMNS, 2018, 3(1), 209-228.10.21042/AMNS.2018.1.00016Search in Google Scholar

[11] 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

[12] 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

[13] Noreen S., Mahmood A., Zagreb Polynomials and Redefined Zagreb Indices for the line graph of Carbon Nanocones, Open J. Math. Anal., 2018, 2(1), 67-76.10.30538/psrp-oma2018.0012Search in Google Scholar

[14] Fath-Tabar G. H., Old and new Zagreb indices of graphs. MATCH Commun, Math. Comput. Chem, 2011, 65(1), 79-84.Search in Google Scholar

[15] Gao W., Asif M., Nazeer W., The Study of Honey Comb Derived Network via Topological Indices, Open J. Math. Anal., 2018, 2(2), 10-26.10.30538/psrp-oma2018.0014Search in Google Scholar

[16] Rehman H. M., Sardar R., Raza A., Computing topological indices of hex board and its line graph, Open J. Math. Sci., 2017, 1, 62-71.10.30538/oms2017.0007Search in Google Scholar

[17] Gao W., Farahani M.R., Shi L., Forgotten topological index of some drug structures, Acta Medica Mediterranea, 2016, 32, 579-585.10.1155/2016/1053183Search in Google Scholar

[18] Wiener H., Structural determination of paraffin boiling points, J. Am. Chem. Soc., 1947, 1(69), 17-20.10.1021/ja01193a005Search in Google Scholar

[19] Deza M., Fowler P.W., Rassat A., Rogers, K.M., Fullerenes as tiling of surfaces, J. Chem. Inf. Comput. Sci., 2000, 40, 550-558.10.1017/CBO9780511721311.004Search in Google Scholar

[20] Randić M., Characterization of molecular branching, J. Am. Chem. Soc., 97(23), 6609-6615.10.1021/ja00856a001Search in Google Scholar

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

[22] Zhou B., Trinajstić N., On a novel connectivity index, J. Math. Chem., 2009, 46, 1252-1270.10.1007/s10910-008-9515-zSearch in Google Scholar

[23] Gutman I, Trinajstić N., Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett., 1972, 17, 535-538.10.1016/0009-2614(72)85099-1Search in Google Scholar

[24] Estrada E., Torres L., Rodriguez L., Gutman I., An atom-bond connectivity index: modelling the enthalpy of formation of alkanes, Indian J. Chem., 1998, 37A, 849-855.Search in Google Scholar

[25] Vukičević D., Furtula B., Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem., 2009, 46, 1369-1376.10.1007/s10910-009-9520-xSearch in Google Scholar

[26] Ghorbani M., Hosseinzadeh M.A., Computing ABC4 index of nanostar dendrimers. Optoelectron, Adv. Mater. Rapid Commun., 2010, 4, 1419-1422.Search in Google Scholar

[27] Graovac A., Ghorbani M., Hosseinzadeh M.A., Computing fifth geometric-arithmetic index for nanostar dendrimers, J. Math. Nanosci., 2011, 1, 33-42.Search in Google Scholar

[28] Hosamani S.M., Computing Sanskruti Index of Certain Nanostructures, J. Appl. Math. Comput., 2016, 54, 425-433.10.1007/s12190-016-1016-9Search in Google Scholar

[29] Furtula B., Graovac A., Vukičević D., Augmented Zagreb index, J. Math. Chem., 2010, 48, 370-380.10.1007/s10910-010-9677-3Search in Google Scholar

[30] Wang D., Huang Y., Liu B., Bounds on augmented Zagreb index, MATCH Commun, Math. Comput. Chem., 2012, 68, 209-216.Search in Google Scholar

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

[32] Hosoya H., On some counting polynomials in chemistry, Discrete Appl. Math., 1988, 19, 239-257.10.1016/0166-218X(88)90017-0Search in Google Scholar

[33] Farrell E.J., An introduction to matching polynomials, J. Combin. Theory Ser., 1979, B 27, 75-86.10.1016/0095-8956(79)90070-4Search in Google Scholar

[34] Chou C.P., Witek H.A., Closed-form formulas for the Zhang-Zhang polynomials of benzenoid structures: chevrons and generalized chevrons, MATCH Commun., Math. Comput. Chem., 2014, 72, 105-124.Search in Google Scholar

[35] Hassani F., Iranmanesh A., Mirzaie S., Schultz and modified Schultz polynomials of C100 fullerene, MATCH Commun, Math. Comput. Chem., 2013, 69, 87-92.Search in Google Scholar

[36] Došlić T., Planar polycyclic graphs and their Tutte polynomials, J. Math. Chem., 2013, 51, 1599-1607.10.1007/s10910-013-0167-2Search in Google Scholar

[37] Dong Y., Xiaofeng L., Yue J., Hao-Bin Z., Bing-Bing J., Hui-Ling M., et al., Thermally conductive phenol formaldehyde composites filled with carbon fillers, Materials Letters, 2014, 118, 212-216.10.1016/j.matlet.2013.12.080Search in Google Scholar

[38] Cheng L., Jizhi Z., Zhao Y., Hua Y., Bin Z., Wei Z., et al., Preparation and characterization of a novel environmentally friendly phenol-formaldehyde adhesive modified with tannin and urea, Int. J. Adhes. Adhes., 2016, 66, 26-32.10.1016/j.ijadhadh.2015.12.004Search in Google Scholar

[39] Foyer G., Chanfi B., Boutevin B., Caillol S., David G., New method for the Synthesis of formaldehyde-free phenolic resins from lignin-based aldehyde precursors, Eur. Polym. J., 2016, 74, 296-309.10.1016/j.eurpolymj.2015.11.036Search in Google Scholar

Received: 2018-12-12
Accepted: 2019-02-21
Published Online: 2019-09-25

© 2019 Maqsood Ahmad et al., published by De Gruyter

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

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