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Computing the Ediz eccentric connectivity index of discrete dynamic structures

  • Hualong Wu , Muhammad Kamran Siddiqui , Bo Zhao EMAIL logo , Jianhou Gan and Wei Gao
Published/Copyright: June 14, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

Abstract

From the earlier studies in physical and chemical sciences, it is found that the physico-chemical characteristics of chemical compounds are internally connected with their molecular structures. As a theoretical basis, it provides a new way of thinking by analyzing the molecular structure of the compounds to understand their physical and chemical properties. In our article, we study the physico-chemical properties of certain molecular structures via computing the Ediz eccentric connectivity index from mathematical standpoint. The results we yielded mainly apply to the techniques of distance and degree computation of mathematical derivation, and the conclusions have guiding significance in physical engineering.

Keywords: theoretical physics; physico-chemical property; molecular graph; Ediz eccentric connectivity index
PACS: 02.10.Ox; 81.05.ub

1 Introduction

In past 40 years, the evidence from many experiments in chemisty, physics, medicine, material and pharmaceutical science implied that the physico-chemical properties of substances heavily depend on their inner molecular structures. For instance, the sum of squares of the vertex degrees of the molecular graph largely influence the structural dependency of total π-electron energy Eπ.

Since the molecular structure is discrete, the graph theory is introduced in computational model, and each molecular structure can be represented as a molecular graph G = (V(G), E(G)), in which V(G) is a set of atoms and E(G) is a set of chemical bonds. A topological index can be considered as a score function f : G → ℝ+ which is introduced as numerical parameters of molecular structures, and it serves as a tool in determining the physico-chemical properties of chemical compounds.

There are lots of serviceable and efficient indices introduced and used in engineering, such as PI index, Wiener index, Zagreb index, atom-bond connectivity index, harmonic index and so on. Some works contributed to determining the distance-based or degree-based indices of special molecular structures. refer to Farahani et al. [1], Jamil et al. [2], Gao et al. [3-6] and Gao and Wang [7, 8] for more details.

For a given vertex u ∈ V(G), the eccentricity ec(u) of u is defined as ec(u) = maxv∈V(G){d(u, v)}. There are several eccentric related indices defined and used in physics, chemisty, material and medicine sciences. Berberler and Berberler [9] determined an explicit formula for the edge eccentric connectivity index of nanothorns. Odabas and Berberler [10] presented explicit formula for the modified eccentric connectivity index of corona molecular graphs. Das and Nadjafi-Arani [11] yielded a lower bound on Szeged index and the eccentric connectivity index in terms of double counting on certain matrix and characterized the extremal molecular structures. Also, they compared the Szeged index and the eccentricity connectivity index of trees. Venkatakrishnan et al. [12] calculated the eccentric connectivity index of several kinds of generalizations of thorn graphs. De et al. [13] presented the exact expressions for the modified eccentric connectivity index and connective eccentric index of V-phenylenic nanotorus. Doslic and Saheli [14] deduced the explicit formulas for eccentric connectivity index for several families of composite graphs, and the results were used to some chemical structures. More results can be found in Ranjini and Lokesha [15], Morgan et al. [16, 17], De [18], Eskender and Vumar [19], Ilić and Gutman [20], Iranmanesh and Hafezieh [21], Dankelmann et al. [22], and Rao and Lakshmi [23].

Ediz introduced the Ediz eccentric connectivity index (denoted by Eζc(G), see Ref. [24] and [25]) of molecular graph G, which is used in various branches of sciences. Specifically, it’s defined as

Eζc(G)=∑v∈V(G)Svec(v),

where S(v) = ∑u∈N(v)d(u).

In this paper, we study the Ediz eccentric connectivity index of some molecular structures. First, we determine the Eζc(G) value of fullerene structures C12n+2, C12n+4 and C18n+10; then, the Ediz eccentric connectivity index of one tetragonal carbon nanocones is discussed; third, the molecular structure of V-phenylenic nanotorus is considered and its corresponding value of Eζc(G) is deduced; at last, we report the Ediz eccentric connectivity index of haphthylenic lattice.

2 Main results and proofs

In this section, we mainly manifest the main results and detail proofs.

2.1 The Ediz eccentric connectivity index of fullerene structures

Fullerenes were discovered in 1985, and are considered as nano molecular structures with zero dimension. Let F be a certain fullerene, and n, m, h, and p be the number of carbon atoms, chemical bonds, hexagons, and pentagons, respectively. By virtue of structure analysis, we can see that each chemical bond lies in two faces and each atom lies in exactly three faces. Using mathematical deduction, we get n=5p+6h3,m=3n2,f = p + h, n − m + f = 2, m = 3h + 30, n = 2h + 20 and p = 12. That is to say, it consists of 12 pentagonal faces, n carbon atoms and n2−10 hexagonal faces. See Prylutskyy et al. [26], Borisova et al. [27], Sugikawa et al. [28], Heumueller et al. [29] and Hendrickson et al. [30], for more details.

The first result in this subsection is manifested, where the discrete structure of C12n+2, C12n+4 and C18n+10 can be referred to Figure 1, Figure 2 and Figure 3, respectively.

Figure 1 The molecular structure of fullerenes C12n+2
Figure 1

The molecular structure of fullerenes C12n+2

Figure 2 The molecular structure of fullerenes C12n+4
Figure 2

The molecular structure of fullerenes C12n+4

Figure 3 The molecular structure of fullerenes C18n+10
Figure 3

The molecular structure of fullerenes C18n+10

Theorem 1

The Ediz eccentric connectivity indices of C12n+2, C12n+4 and C18n+10 are

Eζc(C12n+2)=90n+∑i=1n108n+i,Eζc(C12n+4)=362n+1+∑i=1n+1108n+i,Eζc(C18n+10)=812n+2+632n+3+1352n+1+1352n+2∑i=2n−1(n+i).

Proof

We discuss the above three structures respectively.

  • For fullerenes C12n+2, we have S(v) = 9 for any v ∈ V(C12n+2). Furthermore, the vertex set can be divided into three classes: there are 8 vertices in class 1 with eccentricity 2n; there are 6 vertices in class 2 with eccentricity n; there are 12 vertices for each i with eccentricity n + i where i ∈ {1, 2, · · · , n}. Hence, according to the definition of Ediz eccentric connectivity index, we infer

    Eζc(C12n+2)=6⋅9n+8⋅92n+12∑i=1n9n+i.

  • For fullerenes C12n+4, clearly S(v) = 9 for each v ∈ V(C12n+4). The set V(C12n+4) can be divided into two classes: there are four vertices in class 1 with eccentricity 2n + 1; there are 12 vertices for each i with eccentricity n + i where i ∈ {1, 2, · · · , n + 1}. Thus, in a view of the definition of Ediz eccentric connectivity index, we deduce

    Eζc(C12n+4)=4⋅92n+1+12∑i=1n+19n+i.

  • Similarly, we get S(v) = 9 for any vertex v in C18n+10. Its vertex set can be divided into five classes according to different eccentricities. The eccentricities for vertex in first four classes are 2n, 2n + 1, 2n + 2 and 2n + 3, respectively, and the numbers of vertex are 15, 15, 9, 7, respectively. Furthermore, the last class can be divided into n − 2 subclasses with eccentricities n + i where i ∈ {2, · · · , n − 1}, and each subclass has 18 vertices in it. At last, in terms of the definition of Ediz eccentric connectivity index, we yield

    Eζc(C18n+10)=9⋅92n+2+7⋅92n+3+15⋅92n+1+15⋅92n+18∑i=2n−1n+i9.

Therefore, we complete the proof. □

2.2 The Ediz eccentric connectivity index of one tetragonal carbon nanocones

The molecular structure of tetragonal carbon nanocones CNC4[n] can refer to Kumar and Modan [31] which has 4(n+1)2 atoms and 6n2+10n+4 chemical bonds. The value of eccentricities of all vertices belongs to {2n + 2, · · · , 4n + 2}.

Now, we present the result in this part and the proof is mainly based on the symmetry of molecular structure and graph theory tricks.

Theorem 2

Let CNC4[n] be the tetragonal carbon nanocones.

  1. If n ≡ 1(mod2), then

    Eζc(CNC4[n])=404n+2+∑l=0n−32∑k=ll+172k4n−k−l+1+8(7−k+l)4n−k−l+1+283n+2+9(4n−4)3n+2+36n3n+1+∑l=0n−32∑k=019(3n−k−2l)4n−8l−8.

  2. If n ≡ 0(mod2), then

    Eζc(CNC4[n])=∑l=0n−42∑k=ll+19(8l+8)4n−k−l+8(6+k−l)4n−k−l)+36n+243n+2+∑k=018(2k+5)4n+2−k+∑l=0n−22∑k=019(3n−k−2l+1)4n−8l−4.

Proof

Let CNC4[n]=∪i=1nTi, where Ti is a partition of the molecular graph CNC4[n]. The whole proof can be divided into two cases according to the parity of n.

  • If n ≡ 1(mod2). There are four classes of atoms in each section of Ti: eight atoms of class 1 with maximum eccentricity 4n + 2 and S(v) = 5; there are 8(n − 2l − 2) atoms in class 2 for 0≤l≤n−32 where 4(n − 2l − 2) of them have eccentricity 3n − 2l and other 4(n−2l−2) of them have eccentricity 3n−2l−1. If k = l, then there are eight atoms which satisfy S(v) = 7 and 8k atoms satisfy S(u) = 9. If k = l + 1, then there are eight atoms satisfying S(v) = 6 and 8k atoms satisfy S(u) = 9, and the eccentricity of them is 4n − k − l + 1 for 0≤l≤n−32. At last, there are four atoms which satisfy S(v) = 7, 4n − 4 atoms which satisfy S(v) = 9 and eccentricity is 3n + 2, and 4n atoms which satisfy S(v) = 9 and eccentricity is 3n + 1. Therefore, we get

    Eζc(CNC4[n])=8⋅54n+2+∑l=0n−32∑k=ll+18k⋅94n−k−l+1+8⋅7−k+l4n−k−l+1+4⋅73n+2+(4n−4)⋅93n+2+4n⋅93n+1+∑l=0n−32∑k=01(3n−k−2l)⋅94n−8l−8.

  • If n ≡ 0(mod2). There are four classes of atoms in each section of Ti: there are four atoms with S(v) = 6 and 4n atoms with S(v) = 9 in the first class which have mean eccentric connectivity 3n + 2; there are 8(n − 2l − 1) atoms in the second class for 0≤l≤n−22 where 4(n − 2l − 1) of them have eccentricity 3n − 2l + 1 and other 4(n − 2l − 1) of them have eccentricity 3n − 2l. If k = l, then there are 8l + 8 atoms which S(v) = 9 and eight atoms which satisfy S(u) = 6. If k = l + 1, then there are 8l + 8 atoms satisfying S(v) = 9 and other eight atoms satisfying S(u) = 7, and the eccentricity of them is 4n − k − l for 0≤l≤n−42. At last, there are eight atoms satisfying S(v) = 5 and eight atoms satisfying S(v) = 7 with eccentricity 4n + 2 and 4n + 1, respectively. Therefore, we obtain

    Eζc(CNC4[n])=∑k=018⋅2k+54n+2−k+∑l=0n−42∑k=ll+1(8l+8)⋅94n−k−l+8⋅6+k−l4n−k−l)+4⋅63n+2+4n⋅93n+2+∑l=0n−22∑k=01(3n−k−2l+1)⋅94n−8l−4.

The desired result is proofed. □

2.3 The Ediz eccentric connectivity index of V-phenylenic nanotorus

In this subsection, we focus on the V-phenylenic nanotorus TO[p, q] with p hexagons in each row and q hexagons in each column. As an example, readers can find the molecular structure of TO[4, 5] in Figure 4.

Figure 4 V-phenylenic nanotorus TO [4, 5]
Figure 4

V-phenylenic nanotorus TO [4, 5]

Since TO[p, q] is a cubic discrete structure, we infer that d(u) = 3 (thus S(u) = 9) for each u ∈ V(G), and ec(v) = ec(u) for each pair of atoms (u, v). The result obtained in this part is stated as follows.

Theorem 3

The Ediz eccentric connectivity index of V-phenylenic nanotorus is characterized as follows:

  1. If p ≡ 0(mod2), then

    Eζc(TO[p,q])=108pq4q+p,ifq≥p108pq3p+2q,ifq≤p,

  2. If p ≡ 1(mod2), then

    Eζc(TO[p,q])=108pq4q+p−1,ifq≥p108pq3p+2q−1,ifq≤p,

Proof

We consider the following cases.

  1. p ≡ 0(mod2).

    • If q ≥ p, then ec(v)=4q+p2 for any v ∈ V(G). Then

      REξc(TO[p,q])=108pq4q+p.

    • If q ≤ p, then ec(v)=3p+2q2 for any v ∈ V(G). Then

      REξc(TO[p,q])=108pq3p+2q.

  2. p ≡ 1(mod2).

    • If q ≥ p, then ec(v)=4q+p−12 for any v ∈ V(G). Then

      REξc(TO[p,q])=108pq4q+p−1.

    • If q ≤ p, then ec(v)=3p+2q−12 for any v ∈ V(G). Then

      REξc(TO[p,q])=108pq3p+2q−1.

Thus, we finish the proof of desired result. □

2.4 The Ediz eccentric connectivity index of haphthylenic lattice

The molecular structure of naphthylenic lattice contains the sequence: C6, C6, C4, C6, C6, · · · , C6, C6, C4, C6, C6. As an example, the molecular structure of NP[n, n] is described in Figure 5.

Figure 5 The molecular structure of NP [n, n]
Figure 5

The molecular structure of NP [n, n]

Again, similar as in Theorem 3, we use the discrete method to compute the Ediz eccentric connectivity index of haphthylenic lattice, and our main result in this subsection is listed as follows where we skip the detailed proofs.

Theorem 4

The Ediz eccentric connectivity index of haphthylenic lattice NP[n, n] is listed as follows.

  • If n ≡ 0(mod2), then

    Eζc(NP[n,n])=287n−2+207n−3+167n−4+147n−4+187n−5+167n−5+127n−5+107n−5+367n−6+327n−6+107n−6+547n−7+327n−7+107n−7+727n−8+167n−8+127n−8+5(n−6)7n−8+907n−9+8(n−6)7n−9+147n−9+5(n−6)7n−9+9(n+4)7n−10+8(n+4)7n−10+127n−10+9(2n−2)7n−11+167n−11+⋯9(n−1)9n−4+9(n−2)9n−6+⋯+277n+4+187n+2+97n.

  • If n ≡ 1(mod2), then

    Eζc(NP[n,n])=287n−2+207n−3+167n−4+147n−4+187n−5+167n−5+127n−5+107n−5+367n−6+327n−6+107n−6+547n−7+327n−7+107n−7+727n−8+167n−8+127n−8+5(n−4)7n−8+907n−9+167n−9+147n−9+5(n−7)7n−9+1087n−10+8(n−7)7n−10+127n−10+5(n−7)7n−10+9(n+5)7n−11+8(n−7)7n−11+9(2n−2)7n−12+167n−12+⋯18(2n−3)9n−5+18(2n−5)9n−7+⋯+907n+3+547n+1+187n−1.

Thus, we determined the desired results. □

3 Conclusion

In this paper, by means of discrete dynamic tricks, we determine the Ediz eccentric connectivity index of certain discrete molecular structures which commonly appear in physics, chemistry, engineering, material science, pharmacy and other various branches of science. The results possess very important theoretical guidance value in mentioned applications.

  1. Conflict of interest

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

Acknowledgement

The research is supported by the National Nature Science Fund Project (61562093), and Key Project of Applied Basic Research Program of Yunnan Province (2016FA024).

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Received: 2016-9-13
Accepted: 2016-9-30
Published Online: 2017-6-14

© 2017 H. Wu et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/phys-2017-0039
Keywords for this article
theoretical physics; physico-chemical property; molecular graph; Ediz eccentric connectivity index
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  204. Bohr against Bell: complementarity versus nonlocality
  205. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  206. Is Einsteinian no-signalling violated in Bell tests?
  207. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  208. Bell’s “Theorem”: loopholes vs. conceptual flaws
  209. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  210. Nonrecurrence and Bell-like inequalities
  211. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  212. Three-dimensional computer models of electrospinning systems
  213. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  214. Electric field computation and measurements in the electroporation of inhomogeneous samples
  215. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  216. Modelling of magnetostriction of transformer magnetic core for vibration analysis
  217. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  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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