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Calculating degree-based topological indices of dominating David derived networks

  • Muhammad Saeed Ahmad , Waqas Nazeer , Shin Min Kang EMAIL logo , Muhammad Imran and Wei Gao
Published/Copyright: December 29, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

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)=∑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. χD1n=300081000n2−2963100n+986100,

  2. R′D1n=21n2−21n+223,

  3. AG1D1n=45+213n2−8379100n+2819100,

  4. SK(D1 (n)) = 297n2 − 341n + 121,

  5. SK1D1n=10892n2−13572n+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.

Figure 1 Dominating David derived network of first type D1(2)
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. χD2n=3438100n2−3595100n+1229100,

  2. R′D2n=24n2−763n+9,

  3. AG1D2n=9074100n2−9733100n+3340100,

  4. SK(D2 (n)) = 315n2 − 367n + 131,

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

  6. SK2D2n=25592n2−27752n+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.

Figure 2 Dominating David derived network of 2nd type D2(4)
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. χD3n=66+182n2−4434100n+112,

  2. R′(D3 (n)) = 27n2 − 30n + 11,

  3. AG1D3n=72+272n2−13228100n+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.

Figure 3 Dominating David derived network of 3rd type D3(n)
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 4–9. 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 4

Plots of sum connectivity index

Figure 5 Plots of modified Randić’ index
Figure 5

Plots of modified Randić’ index

Figure 6 Plots of AG1 index
Figure 6

Plots of AG1 index

Figure 7 Plots of SK index
Figure 7

Plots of SK index

Figure 8 Plots of SK1 index
Figure 8

Plots of SK1 index

Figure 9 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.

E-mail:

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

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Received: 2017-9-13
Accepted: 2017-11-11
Published Online: 2017-12-29

© 2017 Muhammad Saeed Ahmad et al.

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

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https://doi.org/10.1515/phys-2017-0126
Keywords for this article
Network; Randić index; degree-based topological index
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  218. Comparison of the fractional power motor with cores made of various magnetic materials
  219. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  220. Dynamics of the line-start reluctance motor with rotor made of SMC material
  221. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  222. Inhomogeneous dielectrics: conformal mapping and finite-element models
  223. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  224. Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
  225. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  226. Detection of inter-turn short-circuit at start-up of induction machine based on torque analysis
  227. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  228. Current superimposition variable flux reluctance motor with 8 salient poles
  229. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  230. Modelling axial vibration in windings of power transformers
  231. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  232. Field analysis & eddy current losses calculation in five-phase tubular actuator
  233. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  234. Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
  235. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  236. Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
  237. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  238. Field-circuit analysis and measurements of a single-phase self-excited induction generator
  239. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  240. A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
  241. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  242. Field-based optimal-design of an electric motor: a new sensitivity formulation
  243. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  244. Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
  245. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  246. Virtual reality as a new trend in mechanical and electrical engineering education
  247. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  248. Holonomicity analysis of electromechanical systems
  249. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  250. An accurate reactive power control study in virtual flux droop control
  251. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  252. Localized probability of improvement for kriging based multi-objective optimization
  253. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  254. Research of influence of open-winding faults on properties of brushless permanent magnets motor
  255. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  256. Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
  257. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  258. Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
  259. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  260. Detection of inter-turn faults in transformer winding using the capacitor discharge method
  261. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  262. A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
  263. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  264. Lamination effects on a 3D model of the magnetic core of power transformers
  265. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  266. Detection of vertical disparity in three-dimensional visualizations
  267. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  268. Calculations of magnetic field in dynamo sheets taking into account their texture
  269. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  270. 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
  271. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  272. Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
  273. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  274. Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
  275. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  276. Speed control of an induction motor by 6-switched 3-level inverter
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