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# Open Physics

### formerly Central European Journal of Physics

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Volume 15, Issue 1

# Computing the Ediz eccentric connectivity index of discrete dynamic structures

Hualong Wu
/ Muhammad Kamran Siddiqui
/ Bo Zhao
• Corresponding author
• School of Information Science and Technology, Yunnan Normal University, Key Laboratory of Educational Information for Nationalities, Ministry of Education, Kunming 650500, China
• Email
• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Jianhou Gan
/ Wei Gao
Published Online: 2017-06-14 | DOI: https://doi.org/10.1515/phys-2017-0039

## 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.

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 uV(G), the eccentricity ec(u) of u is defined as ec(u) = maxvV(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) = ∑uN(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 $n=\frac{5p+6h}{3},\phantom{\rule{thinmathspace}{0ex}}m=\frac{3n}{2},$ f = p + h, nm + 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 $\frac{n}{2}-10$ 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 2

The molecular structure of fullerenes C12n+4

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 vV(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 vV(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 $CN{C}_{4}\left[n\right]={\cup }_{i=1}^{n}{T}_{i},$ 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 $0\le l\le \frac{n-3}{2}$ 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 4nkl + 1 for $0\le l\le \frac{n-3}{2}.$ 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 $0\le l\le \frac{n-2}{2}$ 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 4nkl for $0\le l\le \frac{n-4}{2}.$ 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]

Since TO[p, q] is a cubic discrete structure, we infer that d(u) = 3 (thus S(u) = 9) for each uV(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 qp, then $ec\left(v\right)=\frac{4q+p}{2}$ for any vV(G). Then $REξc(TO[p,q])=108pq4q+p.$

• If qp, then $ec\left(v\right)=\frac{3p+2q}{2}$ for any vV(G). Then $REξc(TO[p,q])=108pq3p+2q.$

2. p ≡ 1(mod2).

• If qp, then $ec\left(v\right)=\frac{4q+p-1}{2}$ for any vV(G). Then $REξc(TO[p,q])=108pq4q+p−1.$

• If qp, then $ec\left(v\right)=\frac{3p+2q-1}{2}$ for any vV(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]

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.

## 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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## About the article

Accepted: 2016-09-30

Published Online: 2017-06-14

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

Citation Information: Open Physics, Volume 15, Issue 1, Pages 354–359, ISSN (Online) 2391-5471,

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