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Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

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Volume 15, Issue 1

Issues

Volume 13 (2015)

Physical-chemical properties studying of molecular structures via topological index calculating

Jianzhang Wu / Mohammad Reza Farahani / Xiao Yu / Wei Gao
Published Online: 2017-05-05 | DOI: https://doi.org/10.1515/phys-2017-0029

Abstract

It’s revealed from the earlier researches that many physical-chemical properties depend heavily on the structure of corresponding moleculars. This fact provides us an approach to measure the physical-chemical characteristics of substances and materials. In our article, we report the eccentricity related indices of certain important molecular structures from mathematical standpoint. The eccentricity version indices of nanostar dendrimers are determined and the reverse eccentric connectivity index for V-phenylenic nanotorus is discussed. The conclusions we obtained mainly use the trick of distance computation and mathematical derivation, and the results can be applied in physics engineering.

Keywords: theoretical physics; Physical-chemical property; molecular graph; topological index; eccentricity

PACS: 02.10.0x

1 Introduction

Many experiments reveal the evidence that the physical-chemical properties of pounds and materials are closely related to their molecular structures. For example, the structure-dependency of total π-electron energy Eπ heavily relies on the sum of squares of the vertex degrees of the molecular graph. Topological indices, as a result, are introduced as numerical parameters of molecular structures, and they play a key role in realizing the physical-chemical properties of substances.

In theoretical computational model, a molecular structure can be represented as a molecular graph G in which each vertex is expressed as an atom and each bound between atoms is denoted as an edge. Then, a topological index can be regarded as a score operator f : G → ℝ+ which maps each molecular graph to a positive real number.

In the past four decades, scientists introduced many indices from the application perspective, such as Zagreb index, Wiener index, sum connectivity index and harmonic index which reflect certain physical-chemical characteristics of molecular structure. There were many contributions to report these distance-based or degree-based indices of special molecular structures (See Farahani et al. [1], Jamil et al. [2], Gao and Wang [3], Gao et al. [47] and Gao and Wang [8, 9] for more details). The notations and terminologies that were used (but undefined in our paper) can be found in Bondy and Murty [10].

For a fixed vertex uV(G), the eccentricity ec(u) of vertex u is defined as the largest distance between u and any other vertex v in G. There are several eccentricity related indices introduced for the engineering purpose.

The first atom-bond connectivity index (in short, ABC index) is introduced by Estrada et al. [11] as ABC(G)=uvE(G)d(u)+d(v)2d(u)d(v),

where d(v) is the degree of vertex v in the molecular graph G. Correspondingly, the fifth atom bond connectivity index is denoted as ABC5(G)=uvE(G)ec(u)+ec(v)2ec(u)ec(v).

The first geometric-arithmetic index (shortly, GA index) rased by Vukić ević and Furtula [12] as GA(G)=uvE(G)2d(u)d(v)d(u)+d(v).

Several results on GA index can refer to Zhou et al. [13], Rodríguez and Sigarreta [14], [15] and [16], Husin et al. [17], Bahrami and Alaeiyan [18], Sigarreta [19], Divnic et al. [20], Das et al. [21], Mahmiani et al. [22], Fath-Tabar et al. [23] and [24], Das et al. [25], Gutman and Furtula [26], Furtula and Gutman [27], and Shabani et al. [28]. The fourth geometric-arithmetic index is defined by Lee et al. [29] which is stated as GA4(G)=uvE(G)2ec(u)ec(v)ec(u)+ec(v).

Moreover, the first and the second multiplicative eccentricity index are defined as Π1(G)=ΠuvE(G)(ec(u)+ec(v)) and Π2(G)=ΠuvE(G)(ec(u)ec(v)), respectively. And, the fourth and sixth Zagreb polynomials are denoted as Zg4(G,x)=uvE(G)xec(u)+ec(v) and Zg6(G,x)=uvE(G)xec(u)ec(v), respectively.

As an important index, the eccentric connectivity index (ECI) of molecular graph G is introduced as ξc(G)=vV(G)ec(v)d(v).

Ranjini and Lokesha [30] determined the eccentric connectivity index of the subdivision graph of tadpole graphs, complete graphs and wheel graphs. Morgan et al. [31] yielded the sharp lower bound on ξc(G) for a given graph order. Furthermore, the tight upper and lower bounds are manifested for trees with given vertex number and diameter. Hua and Das [32] presented the relationship between the eccentric connectivity index and Zagreb indices. De [33] deduced the eccentric connectivity index and polynomial of the thorn graphs. Eskender and Vumar [34] studied the eccentric distance sum and eccentric connectivity index of generalized hierarchical product graphs. In addition, the eccentric connectivity index of F-sum graphs by virtue of certain invariants of the factors are presented as well. Ilić and Gutman [35] proposed that the broom reaches the maximum ξc(G) among trees with fixed maximum vertex degree, and the trees with minimum ξc(G) are also characterized. Iranmanesh and Hafezieh [36] computed the eccentric connectivity index of some special graph families. Dankelmann et al. [37] raised the sharp upper bound for eccentric connectivity index and presented the molecular graphs which asymptotically attain this bound. Morgan et al. [38] presented the tight lower bound of eccentric connectivity index for a tree with given order and diameter. Rao and Lakshmi [39] determined the eccentric connectivity index of phenylenic nanotubes.

Ediz [40] introduced a new distance-based topological index called reverse eccentric connectivity index which is stated as REξc(G)=vV(G)ec(v)S(v),

where S(v)=uN(v)d(u). Nejati and Mehdi [41] obtained the reverse eccentric connectivity index of tetragonal carbon nanocones.

Although there have been several contributions in distance-based and degree-based indices of molecular structures, the study of eccentricity related indices for certain special compounds is still largely limited. On the other hand, as critical and widespread molecular structures, nanostar dendrimers and V-phenylenic nanotorus are widely used in physics, chemistry, biology, medical and material science. For these reasons, we study the exact expressions of eccentricity related indices for nanostar dendrimers and V-phenylenic nanotorus.

2 Main Results and Proofs

In this paper, we mainly study the eccentricity related indices for nanostar dendrimers (Subsection 2.1) and the reverse eccentric connectivity index of V-phenylenic nanotorus (Subsection 2.2).

2.1 Eccentricity related index of nanostar dendrimers

D1[n] is the first class of nanostar dendrimer family in which |V(D1[n])| = 36n – 12 and |E(D1[n])| = 42n – 15, and D2[n] is the second class of nanostar dendrimer family in which |V(D2[n])| = 120 · 2n – 108 and |E(D2[n])| = 140 · 2n – 127. The example of basic structure of D1[n] and D2[n] can refer to Figure 1 and Figure 2, respectively.

D1[1] and D1[2]
Figure 1

D1[1] and D1[2]

D2[1] and D2[2]
Figure 2

D2[1] and D2[2]

Theorem 1

For n = 1, we have GA4(D1[1])=184213+241415+72217+361019,ABC5(D1[1])=91142+31314+152+21710,Π1(D1[1])=139156176196,Π2(D1[1])=429566726906,Zg4(D1[1],x)=9x13+6x15+6x17+6x19,Zg6(D1[1],x)=9x42+6x56+6x72+6x90.

For n ≥ 2, we get GA4(D1[n])=18(3n+3)(3n+4)6n+7+i=1n(32i+1(3n+3i+1)(3n+3i+2)6n+6i+3+32i+1(3n+3i+2)(3n+3i+3)6n+6i+5+32i+1(3n+3i+3)(3n+3i+4)6n+6i+7)+i=1n132i+1(3n+3i+3)(3n+3i+4)6n+6i+7, ABC5(D1[n])=96n+5(3n+3)(3n+4)+i=1n(32i6n+6i+1(3n+3i+1)(3n+3i+2)+32i6n+6i+3(3n+3i+2)(3n+3i+3)+32i6n+6i+5(3n+3i+3)(3n+3i+4))+i=1n132i6n+6i+5(3n+3i+3)(3n+3i+4), Π1(D1[n])=(6n+7)9i=1n((6n+6i+3)32i(6n+6i+5)32i(6n+6i+7)32i)i=1n1(6n+6i+7)32i, Π2(D1[n])=((3n+3)(3n+4))9i=1n(((3n+3i+1)(3n+3i+2))32i((3n+3i+2)(3n+3i+3))32i((3n+3i+3)(3n+3i+4))32i)i=1n1((3n+3i+3)(3n+3i+4))32i, Zg4(D1[n],x)=96n+7+i=1n((32i)6n+6i+3+(32i)6n+6i+5+(32i)6n+6i+7)+i=1n1(32i)6n+6i+7, Zg6(D1[n],x)=9(3n+3)(3n+4)+i=1n((32i)(3n+3i+1)(3n+3i+2)+(32i)(3n+3i+2)(3n+3i+3)+(32i)(3n+3i+3)(3n+3i+4)+i=1n1(32i)(3n+3i+3)(3n+3i+4.

Proof

Since D1[n] is symmetrical, we can mark the vertices some representative symbols which are showed in Figure 1. Next, we only present the detailed proof of GA4 index, and other parts of result can be obtained in the similar way.

By the analysis of molecular structure, the edge set of D1[n] can be divided into six subsets which are described as follows:

  • (u, v): with eccentricities 3n+4 and 3n+3, and there are six edges in this class;

  • (a1, v): with eccentricities 3n + 4 and 3n + 3, and there are three edges in this class;

  • (ai, bi): with eccentricities 3n + 3i+1 and 3n+3i+2, and there are 3 · 2i edges in i-th generation of this class;

  • (bi, ci): with eccentricities 3n+3i+2 and 3n+3i+3, and there are 3 · 2i edges in i-th generation of this class;

  • (ci, di) : with eccentricities 3n+3i+3 and 3n+3i+4, and there are 3 · 2i edges in i-th generation of this class;

  • (ci, ai+1): with eccentricities 3n+3i+3 and 3n+3i+4, and there are 3 · 2i edges in i-th generation of this class.

Thus, by the definition of the fourth GA index, we infer GA4(D1[1])=uvE(D1[1])2ec(u)ec(v)ec(u)+ec(v)=62767+6+32767+6+62787+8+62989+8+629109+10=184213+241415+72217+361019, and for n ≥ 2 GA4(D1[n])=uvE(D1[n])2ec(u)ec(v)ec(u)+ec(v)=62(3n+3)(3n+4)(3n+3)+(3n+4)+32(3n+3)(3n+4)(3n+3)+(3n+4)+i=1n(32i2(3n+3i+1)(3n+3i+2)(3n+3i+1)+(3n+3i+2)+32i2(3n+3i+2)(3n+3i+3)(3n+3i+2)+(3n+3i+3)+32i2(3n+3i+3)(3n+3i+4)(3n+3i+3)+(3n+3i+4))+i=1n132i2(3n+3i+3)(3n+3i+4)(3n+3i+3)+(3n+3i+4)=18(3n+3)(3n+4)6n+7+i=1n(32i2(3n+3i+1)(3n+3i+2)6n+6i+3+32i2(3n+3i+2)(3n+3i+3)6n+6i+5+32i2(3n+3i+3)(3n+3i+4)6n+6i+7)+i=1n132i2(3n+3i+3)(3n+3i+4)6n+6i+7.

Therefore, we get the expected result.  □

Theorem 2

For n = 1, we have GA4(D2[1])=4+161415+24217+241019+811021+163323+323925, ABC5(D2[1]=473+81314+13152+431710+419110+2711+226+42339, Π1(D2[1])=142154172194214234242258,Π2(D2[1])=4925647229041104132414421568, Zg4(D2[1],x)=2x14+4x15+2x17+4x19+4x21+4x23+2x24+8x25, Zg6(D2[1],x)=2x49+4x56+2x72+4x90+4x110+4x132+2x144+8x156.

For n ≥ 2, we get GA4(D2[n])=uvE(D2[n])2ec(u)ec(v)ec(u)+ec(v)=2+8(10n3)(10n2)20n5+4(10n1)(10n2)20n3+i=1n(2i+2(10n+10i10)(10n+10i11)20n20i21+2i+2(10n+10i10)(10n+10i9)20n20i19+2i+2(10n+10i8)(10n+10i9)20n20i17+(2i+22n+2+2n+3)(10n+10i8)(10n+10i7)20n20i15+2i)+i=1n(2i+3(10n+10i5)(10n+10i6)20n20i11+2i+3(10n+10i5)(10n+10i4)20n20i9+2i+3(10n+10i3)(10n+10i4)20n20i7+2i+3(10n+10i3)(10n+10i2)20n20i5+2i+2(10n+10i7)(10n+10i6)20n20i13+2i+2(10n+10i1)(10n+10i2)20n20i3. ABC5(D2[n])=210n320n8+420n7(10n3)(10n2)+220n5(10n1)(10n2)+i=1n(2i+120n+20i23(10n+10i10)(10n+10i11)+2i+120n+20i21(10n+10i10)(10n+10i9)+2i+120n+20i19(10n+10i8)(10n+10i9)+(2i+12n+12n+2)20n+20i17(10n+10i8)(10n+10i7)+2i10n+10i820n+20i18)+i=1n(2i+120n+20i13(10n+10i5)(10n+10i6)+2i+120n+20i11(10n+10i5)(10n+10i4)+2i+220n+20i9(10n+10i3)(10n+10i4)+2i+220n+20i7(10n+10i3)(10n+10i2)+2i+120n+20i15(10n+10i7)(10n+10i6)+2i+120n+20i5(10n+10i1)(10n+10i2)), Π1(D2[n])=(20n6)2(20n5)4(20n3)2i=1n((20n+20i21)2i+1(20n+20i19)2i+1(20n+20i17)2i+1(20n+20i15)(2i+12n+12n+2)(20n+20i16)2i+1)i=1n((20n+20i11)2i+2(20n+20i9)2i+2(20n+20i7)2i+2(20n+20i5)2i+2(20n+20i13)2i+1(20n+20i3)2i+1), Π2(D2[n])=(10n3)4((10n3)(10n2))4((10n1)(10n2))2i=1n(((10n+10i10)(10n+10i11))2i((10n+10i10)(10n+10i9))2i((10n+10i8)(10n+10i9))2i+1((10n+10i8)(10n+10i7))(2i+12n+12n+2)(10n+10i8)2i+1)i=1n(((10n+10i5)(10n+10i6))2i+2((10n+10i5)(10n+10i4))2i+2((10n+10i3)(10n+10i4))2i+2((10n+10i3)(10n+10i2))2i+2((10n+10i7)(10n+10i6))2i+1((10n+10i1)(10n+10i2))2i+1), Zg4(D2[n],x)=2x20n6+4x20n5+2x20n3+i=1n(2i+1x20n+20i21+2i+1x20n+20i19+2i+1x20n+20i17+(2i+12n+12n+2)x20n+20i15+2i+1x20n+20i16+i=1n(2i+2x20n+20i11+2i+2x20n+20i9+2i+2x20n+20i7+2i+2x20n+20i5+2i+1x20n+20i13+2i+1x20n+20i3), Zg6(D2[n],x)=2x(10n3)2+4x(10n3)(10n2)+2x(10n1)(10n2)+i=1n(2i+1x(10n+10i10)(10n+10i11)+2i+1x(10n+10i10)(10n+10i9)+2i+1x(10n+10i8)(10n+10i9)+(2i+12n+12n+2)x(10n+10i8)(10n+10i7)+2i+1x(10n+10i8)(10n+10i8)+i=1n(2i+2x(10n+10i5)(10n+10i6)+2i+2x(10n+10i5)(10n+10i4)+2i+2x(10n+10i3)(10n+10i4)+2i+2x(10n+10i3)(10n+10i2)+2i+1x(10n+10i7)(10n+10i6)+2i+1x(10n+10i1)(10n+10i2)).

Proof

Since D2[n] is also symmetrical, we can mark the vertices some representative symbols which are showed in Figure 2. We only present the detailed proof of GA4 index, and other parts of result can be obtained in the similar way.

By the analysis of molecular structure, the edge set of D2[n] can be divided into 17 subsets which are described as follows:

  • (u, v):with eccentricities 10n – 3 and 10n – 3, and there are two edges in this class;

  • (v, w): with eccentricities 10n – 3 and 10n – 2, and there are four edges in this class;

  • (w, b1):with eccentricities 10n – 2 and 10n – 1, and there are two edges in this class;

  • (ai, bi): with eccentricities 10n+10i–10 and 10n + 10i– 11, and there are 2i edges in i-th generation of this class;

  • (bi, ci): with eccentricities 10n + 10i–11 and 10n +10i– 10, and there are 2i edges in i-th generation of this class;

  • (ci, di): with eccentricities 10n+10i–10 and 10n+10i–9, and there are 2i+1 edges in i-th generation of this class;

  • (di, ei): with eccentricities 10n+10i–9 and 10n+10i–8, and there are 2i+1 edges in i-th generation of this class;

  • (ei, fi): with eccentricities 10n+10i–8 and 10n+10i–7, and there are 2i+1 edges in i-th (in)generation of this class;

  • (en, fn): with eccentricities 10n+10i–8 and 10n+10i–7, and there are 2n+2 edges in this class;

  • (ei, gi): with eccentricities 10n+10i–8 and 10n +10i–8, and there are 2i edges in i-th generation of this class;

  • (ai′, bi′):with eccentricities 10n+10i–5 and 10n+10i– 6, and there are 2i+1 edges in i-th (1 ≤ in –1 and n ≥ 2) generation of this class;

  • (bi′, ci′): with eccentricities 10n+10i–6 and 10n+10i–5, and there are 2i+1 edges in i-th (1 ≤ in –1 and n ≥ 2) generation of this class;

  • (ci′, di′): with eccentricities 10n+10i–5 and 10n+10i–4, and there are 2i+2 edges in i-th (1 ≤ in –1 and n ≥ 2) generation of this class;

  • (di′, ei′): with eccentricities 10n +10i–4 and 10n + 10i– 3, and there are 2i+2 edges in i-th (1 ≤ in – 1 and n ≥ 2) generation of this class;

  • (ei′, fi′): with eccentricities 10n+10i–3 and 10n+10i–2, and there are 2i+2 edges in i-th (1 ≤ in – 1 and n ≥ 2) generation of this class;

  • (fi, bi′):with eccentricities 10n+10i–7 and 10n+10i–6, and there are 2i+1 edges in i-th (1 ≤ in –1 and n ≥ 2) generation of this class;

  • (fi′, bi+1): with eccentricities 10n + 10i–2 and 10n+ 10i – 1, and there are 2i+1 edges in i-th (1 ≤ in – 1 and n ≥ 2) generation of this class.

Thus, in view of the definition of the fourth GA index, we have GA4(D2[1])=uvE(D2[1])2ec(u)ec(v)ec(u)+ec(v)=4+42787+8+22989+8+429109+10+42111011+10+42111211+12+82131213+12=4+161415+24217+241019+811021+163323+323925, and for n ≥ 2 GA4(D2[n])=uvE(D2[n])2ec(u)ec(v)ec(u)+ec(v)=2+8(10n3)(10n2)20n5+4(10n1)(10n2)20n3+i=1n(2i+2(10n+10i10)(10n+10i11)20n20i21+2i+2(10n+10i10)(10n+10i9)20n20i19+2i+2(10n+10i8)(10n+10i9)20n20i17+(2i+22n+2+2n+3)(10n+10i8)(10n+10i7)20n20i15+2i)+i=1n(2i+3(10n+10i5)(10n+10i6)20n20i11+2i+3(10n+10i5)(10n+10i4)20n20i9+2i+3(10n+10i3)(10n+10i4)20n20i7+2i+3(10n+10i3)(10n+10i2)20n20i5+2i+2(10n+10i7)(10n+10i6)20n20i13+2i+2(10n+10i1)(10n+10i2)20n20i3.

Thus, we yield the expected result.  □

2.2 Reverse eccentric connectivity index for V-phenylenic nanotorus

In this subsection, we consider the V-phenylenic nanotorus TO[p, q] (here p is the number of hexagons in each row and q is the number of hexagons in each column) which is a widely used nano structure. For the example of TO[4, 5], please refer to Figure 3.

V-phenylenic nanotorus TO[4, 5]
Figure 3

V-phenylenic nanotorus TO[4, 5]

By the molecular structure analysis, we check that |V(TO[p, q])| = 6pq and |E(TO[p, q])| = 9pq. Furthermore, TO[p, q] is a cubic such that d(v) = 3 for any vV(G) and ec(v) = ec(u) for any two vertices u and v.

Now, we present the main result of this part.

Theorem 3

The reverse eccentric connectivity index for V-phenylenic nanotorus is sated as follows:

  1. If p ≡ 0(mod2), then REξc(TO[p,q])=(4q+p)pq3,ifqp(3p+2q)pq3,ifqp,

  2. If p ≡ 1(mod2), then REξc(TO[p,q])=(4q+p1)pq3,ifqp(3p+2q1)pq3,ifqp,

Proof

Since TO[p, q] is a cubic, we have S(v) = 9 for any vV(G).

The whole proof process can be divided into four parts.

  1. p ≡ 0(mod2) and q ≡ 0(mod2).

    • If qp, then ec(v)=4q+p2 for any vV(G).

      Then REξc(TO[p,q])=(4q+p)pq3.

    • If qp, then ec(v)=3p+2q2 for any vV(G).

      Then REξc(TO[p,q])=(3p+2q)pq3.

  2. p ≡ 1(mod2) and q ≡ 1(mod2).

    • If qp, then ec(v)=4q+p12 for any vV(G).

      Then REξc(TO[p,q])=(4q+p1)pq3.

    • If qp, then ec(v)=3p+2q12 for any vV(G).

      Then REξc(TO[p,q])=(3p+2q1)pq3.

  3. p ≡ 0(mod2) and q ≡ 1(mod2).

    • If qp, then ec(v)=4q+p2 for any vV(G).

      Then REξc(TO[p,q])=(4q+p)pq3.

    • If qp, then ec(v)=3p+2q2 for any vV(G).

      Then REξc(TO[p,q])=(3p+2q)pq3.

  4. p ≡ 1(mod2) and q ≡ 0(mod2).

    • If qp, then ec(v)=4q+p12 for any vV(G).

      Then REξc(TO[p,q])=(4q+p1)pq3.

    • If qp, then ec(v)=3p+2q12 for any vV(G).

      Then REξc(TO[p,q])=(3p+2q1)pq3.

Hence, we get the desired conclusion.  □

3 Conclusion

In this paper, we mainly report the eccentricity related indices for nanostar dendrimers and the reverse eccentric connectivity index of V-phenylenic nanotorus. Since these indices are widely used in the analysis of physical-chemical properties, they possess a promising prospect of application in physical, chemical, medical and material engineering.

Acknowledgement

The authors thank the reviewers for their constructive comments in improving the quality of this paper. This work was supported in part by the National Natural Science Foundation of China [grant number 60903131]; Science and Technology of Jiangsu Province [grant number BE2011173]; and Key Laboratory of Computer Network and Information Integration Founding in Southeast University.

References

  • [1]

    Farahani M.R., Jamil M.K., Imran M., Vertex PIv topological index of titania carbon nanotubes TiO2(m, n), Appl. Math. Nonl. Sc., 2016, 1, 175-182. Google Scholar

  • [2]

    Jamil M.K., Farahani M.R., Imran M., Malik M.A., Computing eccentric version of second Zagreb index of polycyclic aromatic hydrocarbons (PAHk), Appl. Math. Nonl. Sc., 2016, 1, 247-252. Google Scholar

  • [3]

    Gao W., Wang W.F., The eccentric connectivity polynomial of two classes of nanotubes, Chaos, Soliton. Fract., 2016, 89, 290-294. Web of ScienceCrossrefGoogle Scholar

  • [4]

    Gao W., Wang W.F., Farahani M.R., Topological indices study of molecular structure in anticancer drugs, J. Chem., 2016, http://dx.doi.org/10.1155/2016/3216327Web of Science

  • [5]

    Gao W., Farahani M.R., Shi L., Forgotten topological index of some drug structures, Acta Medica Mediterr., 2016, 32, 579-585.Google Scholar

  • [6]

    Gao W., Wang W.F., The fifth geometric arithmetic index of bridge graph and carbon nanocones, J. Differ. Equ. Appl., 2017, http://dx.doi.org/10.1080/10236198.2016.1197214.Web of Science

  • [7]

    Gao W., Wang W.F., Jamil M.K., Farahani M.R., Electron energy studying of molecular structures via forgotten topological index computation, J. Chem., 2016, http://dx.doi.org/10.1155/2016/1053183Web of Science

  • [8]

    Gao W., Wang W.F., Second atom-bond connectivity index of special chemical molecular structures, J. Chem., 2014, http://dx.doi.org/10.1155/2014/906254Web of Science

  • [9]

    Gao W., Wang W.F., The vertex version of weighted wiener number for bicyclic molecular structures, Computational and Mathematical Methods in Medicine, 2015, http://dx.doi.org/10.1155/2015/418106Web of Science

  • [10]

    Bondy J.A., Murty U.S.R., Graph theory, Spring Press, Berlin, 2008. Google Scholar

  • [11]

    Estrada E., Torres L., Rodrí guez L., Gutman I., An atombond connectivity index: modelling the enthalpy of formation of alkanes, Indian J. Chem. A, 1998, 37, 849-855. Google Scholar

  • [12]

    Vukić ević D., Furtula B., Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem., 2009, 46, 1369-1376. Web of ScienceCrossrefGoogle Scholar

  • [13]

    Zhou B., Gutman I., Furtulab B., Dua Z.B., On two types of geometric-rithmetic index, Chem. Phys. Lett., 2009, 482, 153-155. CrossrefGoogle Scholar

  • [14]

    Rodrí guez J.M., Sigarreta J.M., On the geometric-rithmetic index, MATCH Commun. Math. Comput. Chem., 2015, 74, 103-120. Google Scholar

  • [15]

    Rodrí guez J.M., Sigarreta J.M., Spectral properties of geometric-arithmetic index, Appl. Math. Comput., 2016, 277, 142-153. Google Scholar

  • [16]

    Rodrí guez J.M., Sigarreta J.M., Spectral study of the geometric-arithmetic index, MATCH Commun. Math. Comput. Chem., 2015, 74, 121-135. Google Scholar

  • [17]

    Husin M.N., Hasni R., Imran M., Kamarulhaili H., The edge version of geometric arithmetic index of nanotubes and nanotori, Optoelectron. Adv. Mat., 2015, 9, 1292-1300. Google Scholar

  • [18]

    Bahrami A., Alaeiyan M., Fifth geometric-arithmetic index of H-naphtalenic nanosheet [4n, 2m], J. Comput. Theor. Nanos., 2015, 12, 689-690. CrossrefGoogle Scholar

  • [19]

    Sigarreta J.M., Bounds for the geometric-arithmetic index of a graph, Miskolc Math. Notes, 2015, 16, 1199-1212. CrossrefGoogle Scholar

  • [20]

    Divnic T., Milivojevic M., Pavlovic L., Extremal graphs for the geometric-arithmetic index with given minimum degree, Discrete Appl. Math., 2014, 162, 386-390. CrossrefWeb of ScienceGoogle Scholar

  • [21]

    Das, K.C., Trinajstic N., Comparison between geometric-arithmetic indices, Croat. Chem. Acta, 2012, 85, 353-357. CrossrefWeb of ScienceGoogle Scholar

  • [22]

    Mahmiani A., Khormali O., Iranmanesh A., On the edge version of geometric-arithmetic index, Dig. J. Nanomater. Bios., 2012, 7, 411-414. Google Scholar

  • [23]

    Fath-Tabar G.H., Hossein-Zadeh S., Hamzeh A., On the first geometric-arithmetic index of product graphs, Utliltas Mathematica, 2011, 86, 279-287. Google Scholar

  • [24]

    Fath-Tabar G., Furtula B., Gutman I., A new geometric-arithmetic index, J. Math. Chem., 2010, 47, 477-486. CrossrefGoogle Scholar

  • [25]

    Das K.Ch., Gutman I., Furtula B., On the first geometric-arithmetic index of graphs, Discrete Appl. Math., 2011, 159, 2030-2037. Web of ScienceCrossrefGoogle Scholar

  • [26]

    Gutman I., Furtula B., Estimating the second and third geometric-arithmetic indices, Hacet. J. Math. Stat., 2011, 40, 69-76. Google Scholar

  • [27]

    Furtula B., Gutman I., Relation between second and third geometric-arithmetic indices of trees, J. Chemometr., 2011, 25, 87-91. CrossrefWeb of ScienceGoogle Scholar

  • [28]

    Shabani H., Ashrafi A.R., Gutman I., Geometric-arithmetic index: an algebraic approach, Stud. Univ. Babes-Bol., 2010, 55, 107-112. Google Scholar

  • [29]

    Lee D.W., Upper and lower bounds of the fourth geometric-arithmetic index, AKCE Int. J. Graphs Comb., 2013, 10, 69-76. Google Scholar

  • [30]

    Ranjini P.S., Lokesha V., Eccentric connectivity index, hyper and reverse-wiener indices of the subdivision graph, General Mathematics Notes, 2011, 2, 34-46. Google Scholar

  • [31]

    Morgan M.J., Mukwembi S., Swart H.C., On the eccentric connectivity index of a graph, Discrete Math., 2011, 311, 1229-1234. Web of ScienceCrossrefGoogle Scholar

  • [32]

    Hua H., Das K.C., The relationship between the eccentric connectivity index and Zagreb indices, Discrete Appl. Math., 2013, 161, 2480-2491. Web of ScienceCrossrefGoogle Scholar

  • [33]

    De N., On eccentric connectivity index and polynomial of thorn graph, Applied Mathematics, 2012, 3, 931-934. CrossrefGoogle Scholar

  • [34]

    Eskender B., Vumar E., Eccentric connectivity index and eccentric distance sum of some graphs operations, Transactions on Combinatorics, 2013, 2, 103-111. Google Scholar

  • [35]

    Ilić A., Gutman I., Eccentric connectivity index of chemical trees, MATCH Commun. Math. Comput. Chem., 2011, 65, 731-744. Google Scholar

  • [36]

    Iranmanesh M., Hafezieh R., The eccentric connectivity index of some special graphs, Iranian Journal of Mathematical Chemistry, 2011, 2, 61-65. Google Scholar

  • [37]

    Dankelmann P., Morgan M.J., Mukwembi S., Swart H.C., On the eccentric connectivity index and wiener index of a graph, Quaest. Math., 2014, 37, 39-47. CrossrefWeb of ScienceGoogle Scholar

  • [38]

    Morgan M.J., Mukwembi S., Swart H.C., A lower bound on the eccentric connectivity index of a graph, Discrete Appl. Math., 2012, 160, 248-258. CrossrefWeb of ScienceGoogle Scholar

  • [39]

    Rao N.P., Lakshmi K., Eccentric connectivity index of V-phenylenic nanotubes, Dig. J. Nanomater. Bios., 2010, 6, 81-87. Google Scholar

  • [40]

    Ediz S., Reverse eccentric connectivity index, Optoelectron. Adv. Mat., 2012, 6, 664-667. Google Scholar

  • [41]

    Nejati A., Mehdi A., On reverse eccentric connectivity index of one tetragonal carbon nanocones, Journal of Theoretical and Computational Science, 2014, http://dx.doi.org/10.4172/jtco.1000115

About the article

Received: 2016-08-14

Accepted: 2016-09-23

Published Online: 2017-05-05


Conflicts of interest: 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 261–269, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0029.

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© 2017 J. Wu et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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