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Open Physics

formerly Central European Journal of Physics

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Volume 15, Issue 1

Issues

Volume 13 (2015)

Calculating degree-based topological indices of dominating David derived networks

Muhammad Saeed Ahmad / 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
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/ Muhammad Imran / 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].

Keywords: Network; Randić index; degree-based topological index

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)=uvE(G)(du+dv),

and M2(G)=uvE(G)(du×dv).

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

and modified Randić index is defined as RG=uvEG1maxdu,dv.

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

AG1(G)=uvE(G)du+dv2dudv,SK(G)=uvE(G)du+dv2,SK1(G)=uvE(G)du×dv2,SK2(G)=uvE(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. χD1n=300081000n22963100n+986100,

  2. RD1n=21n221n+223,

  3. AG1D1n=45+213n28379100n+2819100,

  4. SK(D1 (n)) = 297n2 − 341n + 121,

  5. SK1D1n=10892n213572n+5012,

  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.

Dominating David derived network of first type D1(2)
Figure 1

Dominating David derived network of first type D1(2)

E1D1n=uvED1n:du=2,dv=2,E2D1n=uvED1n:du=2,dv=3,E3D1n=uvED1n:du=2,dv=4,E4D1n=uvED1n:du=3,dv=3,E5D1n=uvED1n:du=3,dv=4,E6D1n=uvED1n:du=4,dv=4.

It can be observed from Figure 1, that

E1D1n=4n,E2D1n=4n4,E3D1n=28n16,E4D1n=9n213n+5,E5D1n=36n256n+24,E6D1n=36n252n+20.

Now,

χD1(n)=uvE(D1(n))1du+dv=E1D1n12+2+E2D1n12+3+E3D1n12+4+E4D1n13+3+E5D1n13+4+E6D1n14+4=4n4+4n45+28n166+9n213n+56+36n256n+247+36n252n+208=300081000n22963100n+986100. RD1(n)=uvED1(n)1maxdu,dv=uvE1D1n1maxdu,dv+uvE2D1n1maxdu,dv+uvE3D1n1maxdu,dv+uvE4D1n1maxdu,dv+uvE5D1n1maxdu,dv+uvE6D1n1maxdu,dv=E1D1n1max2,2+E2D1n1max2,3+E3D1n1max2,4+E4D1n1max3,3+E5D1n1max3,4+E6D1n1max4,4=4n2+4n43+28n164+9n213n+53+36n256n+244+36n252n+204=21n221n+223. AG1D1n=uvED1(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+10n6106+42n2242+9n213n+5+63n2398n3+423+36n252n+20=45+213n28379100n+2819100. SKD1n=uvED1ndu+dv2=E1D1n2+22+E2D1n2+32+E3D1n2+42+E4D1n3+32+E5D1n3+42+E6D1n4+42=4n2+22+4n42+32+28n162+42+9n213n+53+32+36n256n+243+42+36n252n+204+42=297n2341n+121. SK1D1n=uvED1ndu.dv2=E1D1n2.22+E2D1n2.32+E3D1n2.42+E4D1n3.32+E5D1n3.42+E6D1n4.42=4n42+4n462+28n164+9n213n+592+36n256n+246+36n252n+208=10892n213572n+5012. SK2D1n=uvED1ndu+dv22=E1D1n2+222+E2D1n2+322+E3D1n2+422E4D1n3+322+E5D1n3+422+E6D1n4+422=4n2+222+4n42+322+28n162+422+9n213n+53+322+36n256n+24×3+422+36n252n+204+422=1098n21342n+490.

Theorem 2.2

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

  1. χD2n=3438100n23595100n+1229100,

  2. RD2n=24n2763n+9,

  3. AG1D2n=9074100n29733100n+3340100,

  4. SK(D2 (n)) = 315n2 − 367n + 131,

  5. SK1 (D2 (n))= 558n2 − 698n + 258,

  6. SK2D2n=25592n227752n+10152.

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.

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

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

E1D2n=uvED2n:du=2,dv=2,E2D2n=uvED2n:du=2,dv=3,E3D2n=uvED2n:du=2,dv=4,E4D2n=uvED2n:du=3,dv=4,E5D2n=uvED2n:du=4,dv=4.

It can be observed from figure 2 that E1D2n=4n,E2D2n=18n222n+6,E3D2n=28n16,E4D2n=36n256n+24,E5D2n=36n252n+20.

Now,

χD2n=uvED2n1du+dv=E1D2n12+2+E2D2n12+3+E3D2n12+4+E4D2n13+4+E5D2n14+4=4n4+18n222n+65+28n166+36n256n+247+36n252n+208=3438100n23595100n+1229100. RD2n=uvED2n1maxdu,dv=E1D2n1max2,2+E2D2n1max2,3+E3D2n1max2,4+E4D2n1max3,4+E5D2n1max4,4=4n2+18n222n+63+28n164+36n256n+244+36n252n+204=24n2763n+9. AG1D2n=uvED2ndu+dv2du.dv=E1D2n2+222.2+E2D2n2+322.3+E3D2n2+422.4+E4D2n3+423.4+E5D2n4+424.4=4n+18n222n+6526+28n16628+36n256n+247212+36n252n+20=9074100n29733100n+3340100. SKD2n=uvED2ndu+dv2=E1D2n2+22+E2D2n2+32+E3D2n2+42+E4D2n3+42+E5D2n4+42=4n2+22+18n222n+62+32+28n16×2+42+36n256n+243+42+36n252n+204+42.=315n2367n+131. SK1D2n=uvED2ndu.dv2=E1D2n2.22+E2D2n2.32+E3D2n2.42+E4D2n3.42+E5D2n4.42=4n42+18n222n+662+28n164+36n256n+246+36n252n+208=558n2698n+258. SK2D2n=uvED2ndu+dv22=E1D2n2+222+E2D2n2+322+E3D2n2+422+E4D2n3+422+E5D2n4+422=4n2+222+18n222n+62+322+28n162+422+36n256n+243+422+36n252n+204+422=25592n227752n+10152.

Theorem 2.3

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

  1. χD3n=66+182n24434100n+112,

  2. R′(D3 (n)) = 27n2 − 30n + 11,

  3. AG1D3n=72+272n213228100n+44,

  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.

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

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

E1D3n=uvED3n:du=2,dv=2,E2D3n=uvED3n:du=2,dv=4,E3D3n=uvED3n:du=4,dv=4.

It can be observed from figure 3 that E1D3n=4n,E2D3n=36n220n,E3D3n=72n2108n+44.

Now, χD3n=uvED3n1du+dv=E1D3n12+2+E2D3n12+4+E3D3n14+4=4n4+36n220n6+72n2108n+448=66+182n24434100n+112. RD3n=uvED3n1maxdu,dv=E1D3n1max2,2+E2D3n1max2,4+E3D3n1max4,4=4n2+36n220n4+72n2108n+444=27n230n+11. AG1D3n=uvED3ndu+dv2du.dv=E1D3n2+222.2+E2D3n2+422.4+E3D3n4+424.4=4n+36n220n628+72n2108n+44=72+272n213228100n+44. SKD3n=uvED3ndu+dv2=E1D3n2+22+E2D3n2+42+E3D3n4+42=4n2+22+36n220n2+32+72n2108n+442+42=396n2484n+176. SK1D3n=uvED3ndu.dv2=E1D3n2.22+E2D3n2.42+E3D3n4.42=4n42+36n220n82+72n2108n+448=720n2936n+352. SK2D3n=uvED3ndu+dv22=E1D3n2+222+E2D3n2+422+E3D3n4+422=4n2+222+20n236n2+422+72n2108n+444+422=1476n21892n+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.

Plots of sum connectivity index
Figure 4

Plots of sum connectivity index

Plots of modified Randić’ index
Figure 5

Plots of modified Randić’ index

Plots of AG1 index
Figure 6

Plots of AG1 index

Plots of SK index
Figure 7

Plots of SK index

Plots of SK1 index
Figure 8

Plots of SK1 index

Plots of SK2 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.

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About the article

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Received: 2017-09-13

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, DOI: https://doi.org/10.1515/phys-2017-0126.

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