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Open Chemistry

formerly Central European Journal of Chemistry


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Topological study of the para-line graphs of certain pentacene via topological indices

Zeeshan Saleem Mufti / Muhammad Faisal Nadeem / Wei Gao / Zaheer Ahmad
Published Online: 2018-11-27 | DOI: https://doi.org/10.1515/chem-2018-0126

Abstract

A topological index is a map from molecular structure to a real number. It is a graph invariant and also used to describe the physio-chemical properties of the molecular structures of certain compounds. In this paper, we have investigated a chemical structure of pentacene. Our paper reflects the work on the following indices:Rα, Mα, χα, ABC, GA, ABC4, GA5, PM1, PM2, M1(G, p)and M1(G, p) of the para-line graph of linear [n]-pentacene and multiple pentacene.

Keywords: Topological Indices; Para-line Graph; Nanostructures; Pentacene

1 Introduction and Preliminaries

Every Chemical compound has physical and chemical properties but some chemicals are biologically active, as well. In fact, many pharmaceutical companies are in search of new antibacterial agents. For this purpose they test thousands of compounds, but biological testing is expensive. To overcome this difficulty some other ways of investigating potential antibiotics involve the correlatiion of structures with biological activities or physical and chemical properties. Topological indices, also called molecular descriptors, can be used to explain physio-chemical properties of molecules. In recent years, many graph invariants have been developed for use in different fields of study, including structural chemistry, theoretical chemistry, environmental chemistry, toxicology, and pharmacology. Due to high industrial demand, researchers are motivated to work on topological indices. As a result of these efforts from researchers, more than 400 topological indices have been discovered. As the geometry of any chemical compound plays a vital role in defining the function of this compound, the topological structures of chemical compounds are being correlated with their chemical properties. Topological indices characterise the physio-chemical properties of molecular structures and are widely used in QSAR/QSPR modeling, chemical documentation, drug design, and database selection and for designing multilinear regression models. Molecular descriptors are generally classified into three types: degree-based indices [1, 2, 3, 4, 5], distance-based indices [6, 7, 8, 9, 10, 11], and indices based on spectrum [12, 13, 14, 15]. Indices based on both(Degrees and distances) are used in research reported in the literature (see [16, 17, 18]).

Pentacene is one of the most popular hydrocarbon semiconductors in chemistry [19]. The name pentacene is a combination of two words: penta, meaning five, and acene, which refers to polycyclic aromatic hydrocarbons with fused benzene rings. The importance of pentacene has dramatically increased in recent years due to its key roles in electronic devices and organic solar cells. As the price of energy rises day by day, researchers continue to look for cheaper sources of energy. Electricity, in particular, is produced in multiple ways, including with solar panels. Solar energy is costly because of the high price of developing traditional silicon-based solar cells, leading to a need to optimize organic solar cells, which can be cheaper than silicon-based cells. Researchers at the Georgia Institute of Technology have discovered a technique to create a lightweight organic solar cell. By using pentacene, researchers were able to transform sunlight into electricity with high efficiency. Pentacene, unlike other materials, is a good semiconductor due its crystalline properties. The importance of pentacene has motivated us to perform topological studies of pentacene and we have obtained some important results which will be useful in the study of chemical and physical properties of pentacene. For other topological studies on pentacene see [20, 21].

Let G be a simple graph (no loops and no multiple edges) including two sets: vertex set V(G) and edge set E(G), respectively. For x ∈ V(G), Nx represents the set of its neighbours in G, and the valency (degree) of vertex xisdx = |Nx| and Sxy∈Nx ?dy. If we insert a vertex between every edge of any graph, then every edge of this graph will be divided into two edges, which yields a subdivision of the entire graph. This process is referred to as the subdivision of graphs, and it is denoted by S(G). A line graph can be constructed from graph G by connecting adjacent edges of G with each other; these edges behave as vertices in a new graph. This new graph is denoted by L(G). The line graph of the subdivision graph is termed the paraline graph of G, which is denoted by L(S(G)) (throughout this paper, we will use G* instead of L(S(G))). On the other hand, we can build G* from G in the following way:

  1. Exchange every vertex x ∈ V(G) by K(x), the complete graph on dx vertices;

  2. There is an edge fitting together a vertex of K(x1) and a vertex of K(x2) in G* if and only if there is an edge fitting together x1 and x2 in G;

  3. For every vertex y of K(x), the valency (degree) of y in G* is the same as the valency(degree) of x in G.

These graphs are popular in structural chemistry. Paraline graphs have garnered less attention from researchers in recent years, however attention is increased these days. One reason for their popularity is the simplicty of construction. To construct any chemical compound, a researcher simply considers the carbon atom skeleton, and uses each atom to represent a vertex and each bond between every two atoms to represent an edge. For example, consider the hydrocarbon ethane (C2H6). The structure of Ethane is represented as molecular structure and molecular graph in Figure 1(a) and (b). A para-line graph of the molecular graph of ethane is shown in Figure 1(c).

(a)molecular structure of Ethane, (b) molecular graph of Ethane, (c) Para-line graph of Ethane.
Figure 1

(a)molecular structure of Ethane, (b) molecular graph of Ethane, (c) Para-line graph of Ethane.

The general Randic connectivity index of G is defined as [12]

Rα(G)=xyE(G)(dxdy)α(1)

Where α represents a real number. If α is −1/2, then R−1/2(G) is said to be the Randic connectivity index of G. Li and Zhao presented the first general Zagreb index [22]:

Mα(G)=xV(G)(dx)α(2)

In 2010, a general sum-connectivity index χα(G) was invented [23]:

χα(G)=xyE(G)(dx+dy)α(3)

The (ABC) index was presented by Estrada [24]. The ABC index of graph G is expressed as

ABC(G)=xyE(G)dx+dy2dxdy(4)

Vukicevic and Furtula announced the geometric arithmetic (GA) index [25]. The geometric-arithmetic index, denoted by GA for graph G, is:

GA(G)=xyE(G)2dxdydx+dy(5)

Another index that belongs to the 4th class of (ABC) index was described by Ghorbani et al. [26] as:

ABC4(G)=xyE(G)Sx+Sy2SxSy(6)

The fifth class of geometric-arithmetic index, denoted by GA5, was presented by Graovac et al. [27] as

GA5(G)=xyE(G)2SxSySx+Sy(7)

In 2013, the Hyper-Zagreb index was introduced as

HM(G)=xyE(G)(dx+dy)2(8)

Ghorbani and Azimi proposed two new types of Zagreb indices of a graph G in 2012. PM1(G) is the first multiple Zagreb index, PM2(G) is the second multiple Zagreb index, and M1(G, p) and M2(G, p) are the first Zagreb polynomial and second Zagreb polynomial, respectively. These factors are defined as:

PM1(G)=xyE(G)(dx+dy)(9)PM2(G)=xyE(G)(dx×dy)(10)M1(G,p)=xyE(G)p(dx+dy)(11)M2(G,p)=xyE(G)p(dx×dy)(12)

Ethical approval: The conducted research is not related to either human or animal use

2 Topological indices of para-line graphs

Ranjini designed the independent relations for an index that was presented by Schultz. These researchers investigated the subdivision of various graphs, including helm, ladder, tadpole and wheel, under the surveillance of the Schultz index [28]. They also investigated the paraline graph of ladder, tadpole and wheel under the Zagrib index [29]. Su and Xu evaluated two indices of paraline graphs of ladder, tadpole and wheel graphs, and named the general sum-connectivity index and co-index in 2015 [30]. Nadeem et.al. computed ABC4 and GA5 index of the para-line graphs of the tadpole, wheel and ladder graphs. They examined some indices, such as Rα, Mα, χα, ABC, GA, ABC4 and GA5 indices of the para-line graph of lattice in 2D− nanotube and nanotorus TUC4C8[p, q].

In our paper, we figured Rα, Mα, χα, ABC, GA, ABC4, GA5, PM1, PM2, M1(G, x), and M1(G, x) indices of the para-line graph of linear [n]-pentacene and multiple pentacene.

2.1 Molecular descriptors of the para-line graph of linear [n]-Pentacene

The molecular graph of linear [n]-pentacene is shown in Figure 2, and it is denoted by Tn. There are 22n vertices and 28n − 2 edges in Tn.

Linear Pentacene
Figure 2

Linear Pentacene

Theorem 2.1

Let G* be the para-line graph of Tn. Then

Mα(G*)=(5n+2)2α+2+3α+1(12n4).

Proof. The graph G* is shown in Figure 3. In G* there are a total of 56n − 4 vertices, among which 20n + 8 vertices are of degree 2 and 36n−12 vertices are of degree because Mα(G*) = (5n + 2)2α+2 + 3α+1(12n − 4).

Paraline Graph of Linear Pentacene.
Figure 3

Paraline Graph of Linear Pentacene.

Theorem 2.2

Let G* be the para-line graph of Tn. Then

  1. Rα(G*) = (10n + 10)4α + (20n − 4)6α + (44n − 16)9α;

  2. χα(G*) = (10n + 10)4α + (20n − 4)5α + (44n − 16)6α;

  3. ABC(G*)=(152+883)n+32323;

  4. GA(G*)=(54+86)n6856.

Proof. The total cardinality of edges of G* is 74n − 10. The edge set E(G*)characterized in the following three disjoint edge sets depends on the degrees of the end vertices, i.e. E(G*) = E1(G*)∪E2(G*)∪E3(G*). The edge partition E1(G*) holds 10n + 10 edges xy, where dx = dy = 2, the edge partition E2(G*) holds 20n − 4 edges xy, where dx = 2 and dy = 3, and the edge partition E3(G*) holds 44n −16 edges xy, where dx = dy = 3. From formulas (1), (3), (4) and (5), we get the desired results.

Theorem 2.3

Let G* be the para-line graph of Tn. Then

  1. ABC4(G*)=(110+42+230+163+14)n+5/26+2535852233015110329

  2. GA5(G*)=(30+801310+288172)n2+169516131096172.

    Proof. If we suppose an edge collection depends on degree sum of neighbours of end vertices, then the set of edges E(G*)can be distributed into seven disjoint sets of edges Ei(G*), i = 4, 5, ?, 10, i.e. E(G*)=i=410Ei(G*).The edge collection E4(G*) holds 10 edges xy, where Sx = Sy = 4, the edge collection E5(G*) holds 4 edges xy, where Sx = 4 and Sy = 5, the edge collection E6(G*) holds 10n − 4 edges xy, where Sx = Sy = 5, the edge collection E7(G*) holds 20n − 4 edges xy, where Sx = 5 and Sy = 8, the edge collection E8(G*) holds 8n edges xy, where Sx = Sy = 8, the edge collection E9(G*) holds 24n − 8 edges xy, where Sx = 8 and Sy = 9, and the edge collection E10(G*) holds 12n − 8 edges xy, where Sx = Sy = 9. From formulas 6 and 7, we obtain the required results.

Theorem 2.4

Let G* be the para-line graph of Tn. Then

  1. HM(G*) = 2124 n − 636

  2. PM1(G*) = 410 n+10 × 520 n−4 × 644 n−16

  3. PM2(G*) = 410 n+10 × 620 n−4 × 944 n−16

Proof. Let G* be the para-line graph of linear pentacene. The edge set E(G*)is distributed in three categories which depends on the degree of end vertices. The first disjoint edge set E1(G*) holds 10n+10 edges xy, where dx = dy = 2. The second disjoint set E2(G*) holds 20n-4 edges xy,where dx = 2, dy = 3.The third disjoint set E3(G*) holds 44n-16 edges xy, where dx = dy = 3. Now, |E1(G)| = e2,2, |E2(G)| = e2,3and |E3(G)| = e3,3. Since,

HM(G*)=xyE(G)(dx+dy)2HM(G*)=xyE1(G)[dx+dy]2+xyE2(G)[dx+dy]2+xyE3(G)[dx+dy]2HM(G*)=16|E1(G)|+25|E2(G)|+36|E3(G)|HM(G*)=16(10n+10)+25(20n4)+36(44n16)

This implies that

HM(G*)=2124n636.

Since,

PM1(G*)=xyE(G)(dx+dy)PM1(G*)xyE1(G)(dx+dy)×xyE2(G)(dx+dy)×xyE3(G)(dx+dy)PM1(G*)=4|E1(G)|×5|E2(G)|×6|E3(G)|=410n+10×520n4×644n16PM1(G*)=410n+10×520n4×644n16.

Now, since

PM2(G*)=xyE(G)(dx×dy)PM2(G*)=xyE1(G)(dx×dy)×xyE2(G)(dx×dy)×xyE3(G)(dx×dy)PM2(G*)=4|E1(G)|×6|E2(G)|×9|E3(G)|=410n+10×620n4×944n16

Theorem 2.5

Let G* be the para-line graph of Tn. Then

  1. M1(G*, p) = (10 n + 10) p4 + (20 n − 4) p5 + (44 n − 16) p6

  2. M2(G*, p) = (10 n + 10) p4 + (20 n − 4) p6 + (44 n − 16) p9

Proof.

M1(G*,p)=xyE(G)p(dx+dy)M1(G*,p)=xyE1(G)p(dx+dy)+xyE2(G)p(dx+dy)+xyE3(G)p(dx+dy)=xyE1(G)p4+xyE2(G)p5+xyE3(G)p6=|E1(G)|p4+|E2(G)|p5+|E3(G)|p6=(10n+10)p4+(20n4)p4+(44n16)p6M2(G*,p)=xyE(G)p(dx×dy)M2(G*,x)=xyE1(G)p(dx×dy)+xyE2(G)p(dx×dy)+xyE3(G)p(dx×dy)=xyE1(G)p4+xyE2(G)p6+xyE3(G)p9=|E1(G)|p4+|E2(G)|p6+|E3(G)|p9=(10n+10)p4+(20n4)p6+(44n16)p9.

Hence proved.

2.2 Molecular descriptors of the para-line graph of multiple Pentacene

The molecular graph of multiple pentacene is shown in Figure 4, and it is denoted by Tm,n. There are 22mn vertices and 33mn − 2m − 5n edges in Tm,n.

Multiple Pentacene.
Figure 4

Multiple Pentacene.

Theorem 2.6

Let G* be the para-line graph of Tm,n. Then

Mα(G*)=(5n+2)2α+2+3α+1(12n4).

Proof. The graph G* is shown in Figure 5. In G* there are total 56n−4 vertices among which 20n+8 degree 2 vertices and 36n − 12 degree 3 vertices. Hence we get Mα(G*) by using formula 2.

Paraline Graph of Multiple Pentacene.
Figure 5

Paraline Graph of Multiple Pentacene.

Theorem 2.7

Let G* be the para-line graph of Tm,n. Then

  1. R(G*) = (10 n + 6 m + 4) 4α + (4 m + 20 n − 8) 6α + (99 mn − 20 m − 55 n + 4) 9α;

  2. χα(G*) = (10 n + 6 m + 4) 4α + (4 m + 20 n − 8) 5α + (99 mn − 20 m − 55 n+ 4) 6α;

  3. ABC(G*)=(1521103)n+(52403)m22+66mn+8/3;

  4. GA(G*)=(45+86)n+(8/5614)m+99mn+81656.

Proof. The subdivision graph S(Tm,n) holds 99mn−10m−25n edges and 198mn−20m−50 vertices in total. The division of the vertices is as follows: The number of vertices of degree two are 8m+20n and the number of vertices of degree three are 66mn −12m−30n. The cardinality edge set E of G* are 99mn−20m−55n +4. The edge set E(G*)splits into three edge categories depends on the degrees of the end vertices, i.e. E(G*) = E1(G*) ∪ E2(G*) ∪ E3(G*). The edge partition E1(G*) holds 10n + 6m + 4 edges xy, where dx = dy = 2, the edge partition E2(G*) holds 4m + 20n − 8 edges xy, where dx = 2 and dy = 3, and the edge partition E3(G*) holds 99mn − 20m − 55n + 4 edges xy, where dx = dy = 3. From formulas (1), (3), (4) and (5), Hence desired result is obtained.

Theorem 2.8

Let G* be the para-line graph of Tm,n. Then

  1. ABC4(G*)=(44m+14+42+110+2301163)n+(1/26+1/5110+2/5351129+2/330)m+268/522/51104/330+809

  2. GA5(G*)=(801310+99m+28817269)n+(26+161310+1695+96172)m192172321310+24.

Proof. If the edge partition under consideration depends on degree sum of neighbours of end vertices then the set of edges E(G*) can be classified into seven disjoint edge sets Ei(G*),i=4,5,...,10,i.e.E(G*)=i=410Ei(G*).The edge partition E4(G*) holds 2m+8 edges xy, where Sx = Sy = 4, the edge partition E5(G*) holds 4m edges xy, where Sx = 4 and Sy = 5, the edge partition E6(G*) holds 10n − 4 edges xy, where Sx = Sy = 5, the edge partition E7(G*) holds 20n + 4m − 8 edges xy, where Sx = 5 and Sy = 8, the edge partition E8(G*) holds 8n edges xy, where Sx = Sy = 8, the edge partition E9(G*) holds 8m+24n −16 edges xy, where Sx = 8 and Sy = 9, and the edge partition E10(G*) holds 99mn−28m−87n + 20 edges xy, where Sx = Sy = 9.From formulas (6) and (7), the required result is obtained.

We find the following indices HM(G), PM1(G) ,PM2 (G), Zagreb polynomials M1(G, x), M2(G, x)by computation for chemical structures of multiple-pentacene.

Theorem 2.9

Let G* be the para-line graph of Tm,n.Then

  1. HM(G*) = 3564 mn − 596 m − 1440 n − 40 ;

  2. PM1(G*) = 410 n+6 m+4 × 54 m+20 n−8 × 699 mn−20 m−55 n+4;

  3. PM2(G*) = 410 n+6 m+4 × 64 m+20 n−8 × 999 mn−20 m−55 n+4;

  4. M1(G*, p) = (10 n + 6 m + 4) p4 +(4 m + 20 n − 8) p5 + (99 mn − 20 m − 55 n + 4) p6 ;

  5. M2(G*, p) = (10 n + 6 m + 4) p4 +(4 m + 20 n − 8) p6 + (99 mn − 20 m − 55 n + 4) p9;

Proof. Let G* be the graph. The edge set E(G*)classified into three edge categories based on degree of end vertices. The first edge partition E1(G) holds 10n + 6m + 4 edges xy,where dx = dy = 2. The second edge partition E2(G) holds 4m + 20n − 8 edges xy,where dx = 2, dy = 3.The third edge partition E3(G) holds 99mn − 20m − 55n + 4 edges xy,where dx = 3, dy = 3.It is simple to observe that |E1(G)| = e2,2 , |E2(G)| = e2,3and |E3(G)| = e3,3. Since,

HM(G*)=xyE(G)(dx+dy)2HM(G*)=xyE1(G)[dx+dy]2+xyE2(G)[dx+dy]2+xyE3(G)[dx+dy]2HM(G*)=16|E1(G)|+25|E2(G)|+36|E3(G)|=16(10n+6m+4)+25(20n+4m8)+36(99mn20m55n+4).

This implies that

HM(G*)=3564mn596m1440n40

Since,

PM1(G*)=xyE(G)(dx+dy)PM1(G*)=xyE1(G)(dx+dy)×xyE2(G)(dx+dy)×xyE3(G)(dx+dy)PM1(G*)=4|E1(G)|×5|E2(G)|×6|E3(G)|=410n+6m+4×520n+4m8×699mn20m55n+4

Now, since

PM2(G*)=xyE(G)(dx×dy)PM2(G*)=xyE1(G)(dx×dy)×xyE2(G)(dx×dy)×xyE3(G)(dx×dy)PM2(G*)=4|E1(G)|×6|E2(G)|×9|E3(G)|=410n+6m+4×620n+4m8×999mn20m55n+4

As,

M1(G*,p)=xyE(G)p(dx+dy)M1(G*,p)=xyE1(G)p(dx+dy)+xyE2(G)p(dx+dy)+xyE3(G)p(dx+dy)=xyE1(G)p4+xyE2(G)p5+xyE1(G)p6=|E1(G)|p4+|E2(G)|p5+|E3(G)|p6=(10n+6m+4)p4+(4m+20n8)p5+(99mn20m55n+4)p6M2(G,p)=xyE(G)?p(dx×dy)M2(G,p)=xyE1(G)p(dx×dy)+xyE2(G)p(dx×dy)+xyE3(G)p(dx×dy)=xyE1(G)p4+xyE2(G)p6+xyE3(G)p9=|E1(G)|p4+|E2(G)|p6+|E3(G)|p9=(10n+6m+4)p4+(4m+20n8)p6+(99mn20m55n+4)p9

Which completes the proof.

3 Conclusion

In our paper, we have figured the indices Rα, Mα, χα, ABC, GA, ABC4, GA5, PM1, PM2, M1(G, x), and M1(G, x) of the para-line graph of linear [n]-pentacene and multiple pentacene. The Randic index is used in cheminformatics for studying organic compounds. This index has a better correlation with physio-chemical properties of alkanes, including boiling points, surface areas, and enthalpies of formation. For the stability of any hydrocarbons such as linear and branched alkanes, the ABC index offers a good model. This index also has a correlation with the stability of strain energy of cycloalkane. For some physiochemical properties, GA index can predict physical properties, chemical reactivity and biological activities better than ABC index. We have studied pentacene theoretically, not experimentally. Our theoretical study on pentacene can be very useful and helpful in understanding the physical properties, chemical reactivity and biological activities of pentacenes. The main results obtained in this paper make it possible to correlate the chemical structure of pentacenes with the large amount of information about their physical features, and these results may be useful in the power industry.

References

  • [1]

    Li X., Shi Y., A survey on the Randic’ index, MATCH Commun. Math. Comput. Chem. 2008, 59(1), 127-156. Google Scholar

  • [2]

    Li X., Shi Y., Wang L., An updated survey on the Randic’ index, in: B.F. I. Gutman (Ed.), Recent Results in the Theory of Randic Index, University of Kragujevac and Faculty of Science Kragujevac, 2008, 9-47. Google Scholar

  • [3]

    Mufti Z.S., Zafar S., Zahid Z., Nadeem M. F., Study of the paraline graphs of certain Benzenoid structures using topological indices. MAGNT Research Report, 2017, 4(3), 110-116. Google Scholar

  • [4]

    Rada J., Cruz R., Vertex-degree-based topological indices over graphs, MATCH Commun. Math. Comput. Chem. 2014, 72, 603-616. Google Scholar

  • [5]

    Hinz A.M,. Parisse D., The average eccentricity of Sierpiński graphs, Graphs and Combinatorics, 2012, 28(5), 671-686. CrossrefGoogle Scholar

  • [6]

    Devillers J,. Balaban A.T., Topological indices and related descriptors in QSAR and QSPAR. CRC Press, 2000. Google Scholar

  • [7]

    Azari M., Iranmanesh A., Harary index of some nano-structures, MATCH Commum. Math. Comput. Chem., 2014, 71, 373-382. Google Scholar

  • [8]

    Feng L., Liu W., Yu G., Li S., The hyper-Wiener index of graphs with given bipartition, Utilitas Math. 2014, 95, 23-32. Google Scholar

  • [9]

    Ali A., Nazeer W., Munir M., Kang S.M., M-Polynomials and Topological Indices Of Zigzag and Rhombic Benzenoid Systems. Open Chemistry, 2018, 16(1), 73-78. Web of ScienceCrossrefGoogle Scholar

  • [10]

    Knor M., Lužar B., Škrekovski R., Gutman I., On Wiener index of common neighborhood graphs, MATCH Commum. Math. Comput. Chem., 2014, 72, 321-332. Google Scholar

  • [11]

    Xu K., Liu M., Das K., Gutman I., Furtula B., A survey on graphs extremal with respect to distance-based topological indices, MATCH Commun. Math. Comput. Chem. 2014, 71, 461-508. Google Scholar

  • [12]

    Schultz H.P., Topological organic chemistry, Graph theory and topological indices of alkanes. Journal of Chemical Information and Computer Sciences, 1989, 29(3), 227-228. Google Scholar

  • [13]

    Xu K., Das K.C., Liu H., Some extremal results on the connective eccentricity index of graphs. Journal of Mathematical Analysis and Applications, 2016, 433(2), 803-817. Web of ScienceCrossrefGoogle Scholar

  • [14]

    Kanna M.R., Jagadeesh R., Topological Indices of Vitamin A., Int. J. Math. And Appl., 2018, 6(1B), 271-279. Google Scholar

  • [15]

    Virk A., Nazeer W., Kang S.M., On Computational Aspects of Bismuth Tri-Iodide., Preprints, 2018, 2018060209, doi: 10.20944/preprints201806.0209.v1. 

  • [16]

    Dehmer M., Emmert-Streib F., Grabner M., A computational approach to construct amultivariate complete graph invariant, Inform. Sci. 2014, 260, 200-208. Google Scholar

  • [17]

    Feng L., Liu W., Ilic’ A., Yu G., The degree distance of unicyclic graphs with given matching number, Graphs Comb., 2013, 29, 449-462. CrossrefWeb of ScienceGoogle Scholar

  • [18]

    Gutman I., Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput. Sci., 1994, 34, 1087-1089. CrossrefGoogle Scholar

  • [19]

    Farahani M. R., Nadeem M. F., Zafar S., , Zahid Z., , Husin M. N., Study of the topological indices of the line graphs of hpantacenic nanotubes. New Front. Chem., 2017, 26(1), 31-38. Google Scholar

  • [20]

    Soleimani N., Mohseni E., Maleki N., Imani N., Some topological indices of the family of nanostructures of polycyclic aromatic hydrocarbons (PAHs). J. Natl. Sci Found. Sri., 2018, 46(1). Web of ScienceGoogle Scholar

  • [21]

    Soleimani N., Nikmehr MJ., Tavallaee HA., Theoretical study of nanostructures using topological indices. Stud. U. Babes-Bol, Che., 2014, 59(4), 139-148. Google Scholar

  • [22]

    Li X., Zhao H., Treeswith the first three smallest and largest generalized topological indices, MATCH Commun. Math. Comput. Chem., 2004, 50, 57-62. Google Scholar

  • [23]

    Zhou B., Trinajstic N., On general sum-connectivity index, J. Math. Chem., 2010, 47, 210-218. CrossrefGoogle 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. Google Scholar

  • [25]

    Vukicevic 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. CrossrefWeb of ScienceGoogle Scholar

  • [26]

    Ghorbani M., Hosseinzadeh M.A., Computing ABC4index of nanostar dendrimers, Optoelectron. Adv. Mater.-Rapid Commun., 2010, 4(9), 1419-1422. 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. Google Scholar

  • [28]

    Ranjini P.S., Lokesha V., Rajan M.A., On the Schultz index of the subdivision graphs, Adv. Stud. Contemp.Math., 2011, 21(3), 279-290. Google Scholar

  • [29]

    Ranjini P.S., Lokesha V., Cangül I.N., On the Zagreb indices of the line graphs of the subdivision graphs, Appl. Math. Comput., 2011, 218, 699-702. Google Scholar

  • [30]

    Su G., Xu L., Topological indices of the line graph of subdivision graphs and their Schur-bounds, Appl. Math. Comput., 2015, 253, 395-401. Google Scholar

About the article

Received: 2018-06-05

Accepted: 2018-08-21

Published Online: 2018-11-27


Conflict of interestAuthors declare no conflict of interest.


Citation Information: Open Chemistry, Volume 16, Issue 1, Pages 1200–1206, ISSN (Online) 2391-5420, DOI: https://doi.org/10.1515/chem-2018-0126.

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© 2018 Zeeshan Saleem Mufti et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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