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# Open Physics

### formerly Central European Journal of Physics

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Volume 15, Issue 1

# Calculating degree-based topological indices of dominating David derived networks

/ Waqas Nazeer
/ Shin Min Kang
• Corresponding author
• Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju, 52828, Korea
• Center for General Education, China Medical University, Taichung 40402, Taiwan
• Email
• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Wei Gao
Published Online: 2017-12-29 | DOI: https://doi.org/10.1515/phys-2017-0126

## Abstract

An important area of applied mathematics is the Chemical reaction network theory. The behavior of real world problems can be modeled by using this theory. Due to applications in theoretical chemistry and biochemistry, it has attracted researchers since its foundation. It also attracts pure mathematicians because it involves interesting mathematical structures. In this report, we compute newly defined topological indices, namely, Arithmetic-Geometric index (AG1 index), SK index, SK1 index, and SK2 index of the dominating David derived networks [1, 2, 3, 4, 5].

PACS: 81.05.-t; 81.07.Nb

## 1 Introduction

Chemical graph theory is a branch of graph theory in which a chemical compound is represented by simple graph called molecular graph in which vertices are atoms of compound and edges are the atomic bounds. A graph is connected if there is at least one connection between its vertices. If a graph does not contain any loop or multiple edge then it is called a network. Between two vertices u and v, the distance is the shortest path between them and is denoted by d(u,v) = dG (u,v) in graph G. For a vertex v of G the “degree” dv is the number of vertices attached with it. The degree and valence in chemistry are closely related with each other. We refer the book [6] for more details. Now a day another emerging field is Cheminformatics, which helps to predict biological activities with the relationship of Structure-property and quantitative structure-activity. Topological indices and Physico-chemical properties are used in prediction of bioactivity if underlined compounds are used in these studies [7, 8, 9, 10, 11].

A number that describe the topology of a graph is called topological index. In 1947, the first and most studied topological index was introduced by Weiner [12]. For more details about this index can be found in [13, 14]. In 1975, Milan Randić′ introduced the Randić′ index [15]. Bollobas et al. [16] and Amic et al. [17] in 1998, working independently defined the generalized Randić′ index. This index was studied by both mathematicians and chemists [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].

The first and the second Zagreb indices are defined as $M1(G)=∑uv∈E(G)(du+dv),$

and $M2(G)=∑uv∈E(G)(du×dv).$

(see [30, 31, 32, 33, 34). Sum connectivity index is defined as $χG=∑uv∈E(G)1du+dv,$

and modified Randić index is defined as $R′G=∑uv∈EG1maxdu,dv.$

Shigehalli and Kanabur [35] introduced following new degree-based topological indices: Arithmetic-Geometric (AG1) index

$AG1(G)=∑uv∈E(G)du+dv2dudv,SK(G)=∑uv∈E(G)du+dv2,SK1(G)=∑uv∈E(G)du×dv2,SK2(G)=∑uv∈E(G)du+dv22.$

For more literature review, see Ref. [36, 37, 38].

In this report, we aim to compute degree-based topological indices of dominating David derived networks of first type, second type and third type. These networks are constructed and studied in [1, 2, 3, 4, 5].

## 2 Main results

In this section we present our computational results.

#### Theorem 2.1

Let D1 (n) be the Dominating David Derived Network of 1st type. Then

1. $\begin{array}{}\chi \left({D}_{1}\left(n\right)\right)=\frac{30008}{1000}{n}^{2}-\frac{2963}{100}n+\frac{986}{100},\end{array}$

2. $\begin{array}{}{R}^{\prime }\left({D}_{1}\left(n\right)\right)=21{n}^{2}-21n+\frac{22}{3},\end{array}$

3. $\begin{array}{}A{G}_{1}\left({D}_{1}\left(n\right)\right)=\left(45+21\sqrt{3}\right){n}^{2}-\frac{8379}{100}n+\frac{2819}{100},\end{array}$

4. SK(D1 (n)) = 297n2 − 341n + 121,

5. $\begin{array}{}S{K}_{1}\left({D}_{1}\left(n\right)\right)=\frac{1089}{2}{n}^{2}-\frac{1357}{2}n+\frac{501}{2},\end{array}$

6. SK2 (D1 (n)) = 1098n2 − 1342n + 490.

#### Proof

In dominating derived network D1 (n) (see Figure 1) there are six type of edges in E(D1 (n)) based on the degree of end vertices, i.e.

Figure 1

Dominating David derived network of first type D1(2)

$E1D1n=uv∈ED1n:du=2,dv=2,E2D1n=uv∈ED1n:du=2,dv=3,E3D1n=uv∈ED1n:du=2,dv=4,E4D1n=uv∈ED1n:du=3,dv=3,E5D1n=uv∈ED1n:du=3,dv=4,E6D1n=uv∈ED1n:du=4,dv=4.$

It can be observed from Figure 1, that

$E1D1n=4n,E2D1n=4n−4,E3D1n=28n−16,E4D1n=9n2−13n+5,E5D1n=36n2−56n+24,E6D1n=36n2−52n+20.$

Now,

$χD1(n)=∑uv∈E(D1(n))1du+dv=E1D1n12+2+E2D1n12+3+E3D1n12+4+E4D1n13+3+E5D1n13+4+E6D1n14+4=4n4+4n−45+28n−166+9n2−13n+56+36n2−56n+247+36n2−52n+208=300081000n2−2963100n+986100.$ $R′D1(n)=∑uv∈ED1(n)1maxdu,dv=∑uv∈E1D1n1maxdu,dv+∑uv∈E2D1n1maxdu,dv+∑uv∈E3D1n1maxdu,dv+∑uv∈E4D1n1maxdu,dv+∑uv∈E5D1n1maxdu,dv+∑uv∈E6D1n1maxdu,dv=E1D1n1max2,2+E2D1n1max2,3+E3D1n1max2,4+E4D1n1max3,3+E5D1n1max3,4+E6D1n1max4,4=4n2+4n−43+28n−164+9n2−13n+53+36n2−56n+244+36n2−52n+204=21n2−21n+223.$ $AG1D1n=∑uv∈ED1(n)du+dv2du.dv=E1D1n2+222.2+E2D1n2+322.3+E3D1(n)2+422.4E4D1n3+323.3++E5D1(n)3+423.4+E6D1n4+424.4=4n+10n6−106+42n2−242+9n2−13n+5+63n23−98n3+423+36n2−52n+20=45+213n2−8379100n+2819100.$ $SKD1n=∑uv∈ED1ndu+dv2=E1D1n2+22+E2D1n2+32+E3D1n2+42+E4D1n3+32+E5D1n3+42+E6D1n4+42=4n2+22+4n−42+32+28n−162+42+9n2−13n+53+32+36n2−56n+243+42+36n2−52n+204+42=297n2−341n+121.$ $SK1D1n=∑uv∈ED1ndu.dv2=E1D1n2.22+E2D1n2.32+E3D1n2.42+E4D1n3.32+E5D1n3.42+E6D1n4.42=4n42+4n−462+28n−164+9n2−13n+592+36n2−56n+246+36n2−52n+208=10892n2−13572n+5012.$ $SK2D1n=∑uv∈ED1ndu+dv22=E1D1n2+222+E2D1n2+322+E3D1n2+422E4D1n3+322+E5D1n3+422+E6D1n4+422=4n2+222+4n−42+322+28n−162+422+9n2−13n+53+322+36n2−56n+24×3+422+36n2−52n+204+422=1098n2−1342n+490.$

#### Theorem 2.2

Let D2 (n) be the dominating David derived network of 2nd type. Then

1. $\begin{array}{}\chi \left({D}_{2}\left(n\right)\right)=\frac{3438}{100}{n}^{2}-\frac{3595}{100}n+\frac{1229}{100},\end{array}$

2. $\begin{array}{}{R}^{\prime }\left({D}_{2}\left(n\right)\right)=24{n}^{2}-\frac{76}{3}n+9,\end{array}$

3. $\begin{array}{}A{G}_{1}\left({D}_{2}\left(n\right)\right)=\frac{9074}{100}{n}^{2}-\frac{9733}{100}n+\frac{3340}{100},\end{array}$

4. SK(D2 (n)) = 315n2 − 367n + 131,

5. SK1 (D2 (n))= 558n2 − 698n + 258,

6. $\begin{array}{}S{K}_{2}\left({D}_{2}\left(n\right)\right)=\frac{2559}{2}{n}^{2}-\frac{2775}{2}n+\frac{1015}{2}.\end{array}$

#### Proof

In dominating derived network D2 (n)(figure 2) there are five type of edges in E(D2 (n)) based on the degree of end vertices. i.e.

Figure 2

Dominating David derived network of 2nd type D2(4)

$E1D2n=uv∈ED2n:du=2,dv=2,E2D2n=uv∈ED2n:du=2,dv=3,E3D2n=uv∈ED2n:du=2,dv=4,E4D2n=uv∈ED2n:du=3,dv=4,E5D2n=uv∈ED2n:du=4,dv=4.$

It can be observed from figure 2 that $E1D2n=4n,E2D2n=18n2−22n+6,E3D2n=28n−16,E4D2n=36n2−56n+24,E5D2n=36n2−52n+20.$

Now,

$χD2n=∑uv∈ED2n1du+dv=E1D2n12+2+E2D2n12+3+E3D2n12+4+E4D2n13+4+E5D2n14+4=4n4+18n2−22n+65+28n−166+36n2−56n+247+36n2−52n+208=3438100n2−3595100n+1229100.$ $R′D2n=∑uv∈ED2n1maxdu,dv=E1D2n1max2,2+E2D2n1max2,3+E3D2n1max2,4+E4D2n1max3,4+E5D2n1max4,4=4n2+18n2−22n+63+28n−164+36n2−56n+244+36n2−52n+204=24n2−763n+9.$ $AG1D2n=∑uv∈ED2ndu+dv2du.dv=E1D2n2+222.2+E2D2n2+322.3+E3D2n2+422.4+E4D2n3+423.4+E5D2n4+424.4=4n+18n2−22n+6526+28n−16628+36n2−56n+247212+36n2−52n+20=9074100n2−9733100n+3340100.$ $SKD2n=∑uv∈ED2ndu+dv2=E1D2n2+22+E2D2n2+32+E3D2n2+42+E4D2n3+42+E5D2n4+42=4n2+22+18n2−22n+62+32+28n−16×2+42+36n2−56n+243+42+36n2−52n+204+42.=315n2−367n+131.$ $SK1D2n=∑uv∈ED2ndu.dv2=E1D2n2.22+E2D2n2.32+E3D2n2.42+E4D2n3.42+E5D2n4.42=4n42+18n2−22n+662+28n−164+36n2−56n+246+36n2−52n+208=558n2−698n+258.$ $SK2D2n=∑uv∈ED2ndu+dv22=E1D2n2+222+E2D2n2+322+E3D2n2+422+E4D2n3+422+E5D2n4+422=4n2+222+18n2−22n+62+322+28n−162+422+36n2−56n+243+422+36n2−52n+204+422=25592n2−27752n+10152.$

#### Theorem 2.3

Let D3 (n) be the dominating David derived network of 3rd type. Then

1. $\begin{array}{}\chi \left({D}_{3}\left(n\right)\right)=\left(6\sqrt{6}+18\sqrt{2}\right){n}^{2}-\frac{4434}{100}n+11\sqrt{2},\end{array}$

2. R′(D3 (n)) = 27n2 − 30n + 11,

3. $\begin{array}{}A{G}_{1}\phantom{\rule{thinmathspace}{0ex}}\left({D}_{3}\phantom{\rule{thinmathspace}{0ex}}\left(n\right)\right)=\phantom{\rule{thinmathspace}{0ex}}\left(72+27\sqrt{2}\right){n}^{2}-\frac{13228}{100}n+44,\end{array}$

4. SK(D3 (n)) = 396n2 − 484n + 176,

5. SK1 (D3 (n)) = 720n2 − 936n + 352,

6. SK2 (D3 (n)) = 1476n2 − 1892n + 704.

#### Proof

In dominating derived network D3 (n) (Figure 3) there are three type of edges in E(D3 (n)) based on the degree of end vertices. i.e.

Figure 3

Dominating David derived network of 3rd type D3(n)

$E1D3n=uv∈ED3n:du=2,dv=2,E2D3n=uv∈ED3n:du=2,dv=4,E3D3n=uv∈ED3n:du=4,dv=4.$

It can be observed from figure 3 that $E1D3n=4n,E2D3n=36n2−20n,E3D3n=72n2−108n+44.$

Now, $χD3n=∑uv∈ED3n1du+dv=E1D3n12+2+E2D3n12+4+E3D3n14+4=4n4+36n2−20n6+72n2−108n+448=66+182n2−4434100n+112.$ $R′D3n=∑uv∈ED3n1maxdu,dv=E1D3n1max2,2+E2D3n1max2,4+E3D3n1max4,4=4n2+36n2−20n4+72n2−108n+444=27n2−30n+11.$ $AG1D3n=∑uv∈ED3ndu+dv2du.dv=E1D3n2+222.2+E2D3n2+422.4+E3D3n4+424.4=4n+36n2−20n628+72n2−108n+44=72+272n2−13228100n+44.$ $SKD3n=∑uv∈ED3ndu+dv2=E1D3n2+22+E2D3n2+42+E3D3n4+42=4n2+22+36n2−20n2+32+72n2−108n+442+42=396n2−484n+176.$ $SK1D3n=∑uv∈ED3ndu.dv2=E1D3n2.22+E2D3n2.42+E3D3n4.42=4n42+36n2−20n82+72n2−108n+448=720n2−936n+352.$ $SK2D3n=∑uv∈ED3ndu+dv22=E1D3n2+222+E2D3n2+422+E3D3n4+422=4n2+222+20n2−36n2+422+72n2−108n+444+422=1476n2−1892n+704.$

## 3 Graphical comparison

In this section we give graphical comparison of our results Figures 49. Turquoise color is for dominating David derived networks of first type, lime color is for dominating David derived network of second type and purple color is for dominating David derived network of third type.

Figure 4

Plots of sum connectivity index

Figure 5

Plots of modified Randić’ index

Figure 6

Plots of AG1 index

Figure 7

Plots of SK index

Figure 8

Plots of SK1 index

Figure 9

Plots of SK2 index

## 4 Conclusions

In the present report, we computed seven degree-based topological indices of dominating David derived networks of first, second and third type. We compare our results geometrically by plotting computed degree-based indices. We believe that our results play a vital rule in preparation of new drugs.

## References

• [1]

Imran M., Baig, A.Q., Ali, H. On topological properties of dominating David derived networks, Canad. J. of Chem., 2015, 94(2), 137-148. Google Scholar

• [2]

Star of David [online]. Available at http://Wikipedia.org/wiki/starofDavid

• [3]

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

• [4]

Trinajstic N., Chemical graph theory, CRC press, 1992 Google Scholar

• [5]

Simonraj F., George A., Graph-Hoc 2012, 4, 11,

• [6]

West D.B., Introduction to graph theory (Vol. 2), Upper Saddle River, Prentice Hall, 2001 Google Scholar

• [7]

Rücker G., Rücker C., On topological indices, boiling points, and cycloalkanes, J. of Chem. Inf. and Comp. Sci.,1999, 39(5), 788-802.

• [8]

Klavzar S., Gutman I., A comparison of the Schultz molecular topological index with the Wiener index, J. of Chem. inf. and comp. sci., 1996, 36(5), 1001-1003.

• [9]

Brückler F.M., Došliæ T., Graovac A., Gutman I., On a class of distance-based molecular structure descriptors, Chem. Phys. Lett., 2011, 503(4), 336-338.

• [10]

Deng H., Yang J., Xia F., A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes, Comp. & Math. with Applic., 2011, 61(10), 3017-3023.

• [11]

Zhang H., Zhang F., The Clar covering polynomial of hexagonal systems I, Discr. Appl. Math.,1996, 69(1), 147-167.

• [12]

Wiener H., Structural determination of paraffin boiling points, J. Amer. Chem. Soc., 1947, 69(1), 17-20.

• [13]

Dobrynin A.A., Entringer R., Gutman I., Wiener index of trees: theory and applications, Acta Applic. Math., 2001, 66(3), 211-249.

• [14]

Gutman I., Polansky O.E., Mathematical concepts in organic chemistry, Springer Science & Business Media, 2012. Google Scholar

• [15]

Randić M., Characterization of molecular branching, J. Amer.Chem. Soc.,1975, 97(23), 6609-6615.

• [16]

Bollobás B., Erdõs P., Graphs of extremal weights, Ars Combinatoria, 1998, 50, 225-233. Google Scholar

• [17]

Amic D., Bešlo D., Lucic B., Nikolic S., Trinajstic N., The vertex connectivity index revisited, J. of Chem. Inf. and Comp. Sci., 1998, 38(5), 819-822.

• [18]

Hu Y., Li X., Shi Y., Xu T., Gutman I., On molecular graphs with smallest and greatest zeroth-order general Randić index, MATCH Commun. Math. Comput. Chem, 2005, 54(2), 425-434. Google Scholar

• [19]

Caporossi G., Gutman I., Hansen P., Pavloviæ L., Graphs with maximum connectivity index, Comp. Biol. and Chem., 2003, 27(1), 85-90.

• [20]

Li X., Gutman I., Mathematical Chemistry Monographs No. 1, Kragujevac, 2006. Google Scholar

• [21]

Hall L.H., Kier L.B., Molecular connectivity in chemistry and drug research, 1976 Google Scholar

• [22]

Kier L.B., Hall L.H., Molecular connectivity in structure-activity analysis, Research Studies, 1986. Google Scholar

• [23]

Li X., Gutman I., Randić M., Mathematical aspects of Randić-type molecular structure descriptors, University, Faculty of Science, 2006. Google Scholar

• [24]

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

• [25]

Randić M., The connectivity index 25 years after, J. Mol. Graph. Mod., 2001, 20(1), 19-35.

• [26]

Gutman I., Furtula B., (Eds.) Recent results in the theory of Randic index, University, Faculty of Science, 2008. Google Scholar

• [27]

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

• [28]

Gutmann I., (Ed.) Recent Results in the Theory of Randić Index, Kragujevac University, 2008. Google Scholar

• [29]

Nikolic S., Kovaèeviæ G., Milièevic A., Trinajstic N., The Zagreb indices 30 years after, Croatica Chemica Acta, 2003, 76(2), 113-124.Google Scholar

• [30]

Gutman I., Das K.C., The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem, 2004, 50, 83-92. Google Scholar

• [31]

Das K. C., Gutman I., Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem, 2004, 52(1), 3-1. Google Scholar

• [32]

Trinajstic N., Nikolic S., Milièevic A., Gutman I., About the Zagreb Indices, Kemija u industriji, 2010, 59(12), 577-589. Google Scholar

• [33]

Vukièevic D., Graovac A., Valence connectivity versus Randić, Zagreb and modified Zagreb index: A linear algorithm to check discriminative properties of indices in acyclic molecular graphs, Croatica Chemica Acta, 2004, 77(3), 501-508. Google Scholar

• [34]

Milièevic A., Nikolic S., Trinajstic N., On reformulated Zagreb indices, Mol. Div., 2004, 8(4), 393-399.

• [35]

Shigehalli V.S., Kanabur R., Computation of New Degree-Based Topological Indices of Graphene, 2016, 2016, 4341919 Google Scholar

• [36]

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.

• [37]

Rehman H.M., Sardar R., Raza A., Computing Topological Indices of Hex Board and its Line Graph, Open J. Math. Sci., 2017, 1, 1, 62-71.

• [38]

Sardar M. S., Pan x. R., Gao W., Farahani M. R., Computing Sanskruti Index of Titania Nanotubes, Open J. Math. Sci., 2017, 1, 1, 126-130.

## About the article

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Accepted: 2017-11-11

Published Online: 2017-12-29

Conflict of Interest: The authors declare that there is no conflict of interest regarding the publication of this paper.

Citation Information: Open Physics, Volume 15, Issue 1, Pages 1015–1021, ISSN (Online) 2391-5471,

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