Distance-based Indices Computation of Symmetry Molecular Structures

Author(s): Li Yan1, Mohammad Reza Farahani2, Wei Gao3
1School of Engineer, Honghe University, Mengzi 661100, China.
2Department of Applied Mathematics, Iran University of Science and Technology, Tehran 16844, Iran.
3School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China.
Copyright © Li Yan, Mohammad Reza Farahani, Wei Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Most of molecular structures have symmetrical characteristics. It inspires us to calculate the topological indices by means of group theory. In this paper, we present the formulations for computing the several distance-based topological indices using group theory. We solve some examples as applications of our results.

Keywords: molecular graph, group theory, generalized Harary index.

1. Introduction

In early years, many chemical experiments showed the evidence that the biochemical properties of chemical compounds, materials and drugs are closely related to their molecular structures. As a result, topological indices are introduced as numerical parameters of molecular graph, which play a vital role in understanding the properties of chemical compounds and are applied in disciplines such as chemistry, physics and medicine science.

In chemical graph theory, a molecular structure is expressed as a molecular graph G in which atoms are taken as vertices and chemical bonds are taken as edges. A topological index can be considered as a function f:GR+. In the past 40 years, scholars introduced many topological indices, such as Wiener index, Zagreb index, harmonic index, sum connectivity index, etc which reflect certain structural characteristics of organic molecules. There were many works contributing to report these distance-based or degree-based indices of special molecular structures (See Farahani et al. [1], Jamil et al. [2], Gao and Farahani [3], Gao et al. [4, 5, 6] and Gao and Wang [7, 8, 9] for details). The notation and terminology that were used but were undefined in this paper can be found in [10].

One of oldest indices, the Wiener index was defined as the sum of distance for all pair of vertices, W(G)={u,v}V(G)d(u,v). The modified Wiener index was introduced by Nikolić et al. [11] as the extension of the Wiener index which was defined as Wλ(G)={u,v}V(G)dλ(u,v). Several conclusions on modified Wiener index can be referred to Vukičević and Žerovnik [12], Vukičević and Gutman [13], Lim [14], Gorse and Žerovnik [15], Vukičević and Graovac [16], and Gutman et al. [17].

Moreover, the hyper-Wiener index and λ-modified hyper-Wiener index are defined as WW(G)={u,v}V(G)12(d(u,v)+d2(u,v)) and WWλ(G)={u,v}V(G)12(dλ(u,v)+d2λ(u,v)), respectively. Some important contributions on hyper-Wiener index can be found in Gutman [18], Gutman and Furtula [19], Eliasi and Taeri [20, 21], Iranmanesh et al. [22], Yazdani and Bahrami [23], Behtoei et al. [24], Mansour and Schork [25], Heydari [26], Ashrafi et al. [27], and Heydari [28].

The Harary index was introduced independently by Plavšić et al. [29] and Ivanciuc et al. [30] in 1993, as H(G)={u,v}V(G)1d(u,v). Its corresponding Harary polynomial can be defined as H(G,x)={u,v}V(G)1d(u,v)xd(u,v). The second and third Harary indices are defined as H1(G)={u,v}V(G)1d(u,v)+1, H2(G)={u,v}V(G)1d(u,v)+2. More generally, the generalized Harary index was introduced by Das et al. [31] which is defined as Ht(G)={u,v}V(G)1d(u,v)+t, where tN is a non-negative integer. Hence, Harary index is a special case of generalized Harary index when t=0.

One topological index related to Wiener index is the reciprocal complementary Wiener (RCW) index which is defined by Zhou et al. [32] and can be defined as RCW(G)={u,v}V(G)1D(G)+1d(u,v), where D(G) is the diameter of molecular graph G. In what follows, we always denote D(G) as the diameter of molecular graph G.

Furthermore, the multiplicative version of the Wiener index was defined by Gutman et al. [33, 34] as π(G)={u,v}V(G)d(u,v). The logarithm of multiplicative Wiener index was defined as Π(G)=ln2{u,v}V(G)d(u,v).

So far, many mathematical approaches are given to calculate different topological indices and received good results. Since most of the chemical compounds have symmetric structures. It inspires us to consider the computation of topological indices by using group theory. We use the automorphism groups and its orbits to simplify the computation of molecular graphs for some distance-based indices.

2. Main results and proofs

To discuss the symmetry molecular structures, we should first introduce symmetry operations which are defined as operations that move a fixed molecule structure from a previous condition to another, and any two states can’t be differentiated from each other. Obviously, all the symmetry operations on a molecular structure constitute a group which is called the point group of the molecular structure.

When an element p of point group P (i.e., a symmetry operation on the molecular structure) operates on the molecular graph, it provides the vertices of the molecular graph a permutation. We denote p(v) as the image of vertex v under the operation p. If there exists a pP that satisfies p(v)=u for two vertices v and u, then we define an equivalence binary relation denoted by vu. By means of this equivalent relation, the vertex set is divided into several equivalence classes: Θ1,,Θr. Each Θi can be called an orbit of point group P, and the number of vertices of each orbit |Θi| is called the length of the orbit Θi. By the knowledge of group theory, we know that if vΘi then |Θi|=|P||Pv|, where Pv={p|pP,p(v)=v}. The group called transitive if it has only one orbit, and it is called intransitive otherwise. Moreover, we can define the orbits of subgroup H in similar way which could also be either transitive or intransitive.

Set
  1. Wλ(v,G)=uV(G)dλ(u,v),
  2. WW(v,G)=12uV(G)(d(u,v)+d2(u,v)),
  3. WWλ(v,G)=12uV(G)(dλ(u,v)+d2λ(u,v)),
  4. H(v,G)=uV(G)1d(u,v),
  5. H(v,G,x)=uV(G)1d(u,v)xd(u,v),
  6. H1(v,G)=uV(G)1d(u,v)+1,
  7. H2(v,G)=uV(G)1d(u,v)+2,
  8. Ht(v,G)=uV(G)1d(u,v)+t,
  9. RCW(v,G)=uV(G)1D(G)+1d(u,v),
  10. π(v,G)=uV(G)d(u,v).
Hence, we have
  1. Wλ(G)=12vV(G)Wλ(v,G)(1)
  2. WW(G)=12vV(G)WW(v,G),
  3. WWλ(G)=12vV(G)WWλ(v,G),
  4. H(G)=12vV(G)H(v,G),
  5. H(G,x)=12vV(G)H(v,G,x),
  6. H1(G)=12vV(G)H1(v,G),
  7. H2(G)=12vV(G)H2(v,G),
  8. Ht(G)=12vV(G)Ht(v,G),
  9. RCW(G)=12vV(G)RCW(v,G),
  10. π(G)=vV(G)π(v,G),
  11. Π(G)=ln2vV(G)π(v,G).
Following theorem is about the calculation of different topological indices when the point group is not necessarily transitive.

Theroem 2.1. Let HPG be a subgroup of PG, and Θ1,Θ2,,Θr are the orbits of H and uiΘi, i=1,2,,r. Then, we have

  1. Wλ(G)=i=1rj=i+1r|Θi|uΘjdλ(u,ui)+12i=1r|Θi|vΘidλ(v,ui),
  2. WW(G)=12i=1rj=i+1r|Θi|uΘj(d(u,ui)+d2(u,ui))+12i=1r|Θi|vΘi(d(v,ui)+d2(v,ui)),
  3. WWλ(G)=12i=1rj=i+1r|Θi|uΘj(dλ(u,ui)+d2λ(u,ui))+12i=1r|Θi|vΘi(dλ(v,ui)+d2λ(v,ui)),
  4. H(G)=i=1rj=i+1r|Θi|uΘj1d(u,ui)+12i=1r|Θi|vΘi1d(v,ui),
  5. H(G,x)=i=1rj=i+1r|Θi|uΘj1d(u,ui)xd(u,ui)+12i=1r|Θi|vΘi1d(v,ui)xd(v,ui),
  6. Ht(G)=i=1rj=i+1r|Θi|uΘj1d(u,ui)+t+12i=1r|Θi|vΘi1d(v,ui)+t,
  7. RCW(G)=i=1rj=i+1r|Θi|uΘj1D(G)+1d(u,ui)+12i=1r|Θi|vΘi1D(G)+1d(v,ui),
  8. π(G)=i=1rj=i+1ruΘjd|Θi|(u,ui)×i=1rvΘid|Θi|(v,ui),
  9. Π(G)=ln2i=1rj=i+1ruΘjd|Θi|(u,ui)×i=1rvΘid|Θi|(v,ui).

Proof. We only prove for Wλ(G). The remaining cases can be proved in similar fashion. Since Wλ(u,G) is equal to the sum of all vertices in the same orbit, we infer wV(G)Wλ(w,G)=i=1rwΘiWλ(w,G)=i=1r|Θi|Wλ(ui,G)=i=1r|Θi|j=1ryΘidλ(ui,y)=i=1rj=1r|Θi|yΘidλ(ui,y)=2i=1rj=i+1r|Θi|yΘidλ(y,ui)+i=1r|Θi|zΘidλ(z,ui). Hence, in terms of (1), Wλ(G)=12wV(G)Wλ(w,G)=i=1rj=i+1r|Θi|uΘjdλ(u,ui)+12i=1r|Θi|vΘidλ(v,ui). Hence, the desired result is obtained.

The next result is about the computation of topological indices when the point group of the molecular graph is transitive.

Lemma 2.2. If the point group PG of the molecular graph is transitive. Then for any vV(G), we have

  1. Wλ(G)=|V(G)|2Wλ(v,G),
  2. WW(G)=|V(G)|4WW(v,G),
  3. WWλ(G)=|V(G)|4WWλ(v,G),
  4. H(G)=|V(G)|2H(v,G),
  5. H(G,x)=|V(G)|2H(v,G,x),
  6. Ht(G)=|V(G)|2Ht(v,G),
  7. RCW(G)=|V(G)|2RCW(v,G),
  8. π(G)=π|V(G)|(v,G),
  9. Π(G)=ln2π|V(G)|(v,G).

In terms of Lemma 2.2, to calculate the distance-based topological indices of the molecular graph with transitive point group, we only need to choose any vertex vV(G) and calculate the distances between v and uV(G){v}. Take a subgroup H of PG, which is not necessarily transitive even if PG is transitive. Now, the vertex set V(G) can be divided into orbits of H such that Θ1,Θ2,,Θr with (|Θ1||Θ2||Θr|.

Theorem 2.3. Let viΘi, i=1,2,,r. Then,

  1. Wλ(v1,G)=1|Θ1|uΘ1i=1r|Θi|dλ(u,vi),
  2. WW(v1,G)=1|Θ1|uΘ1i=1r|Θi|2(d(u,vi)+d2(u,vi)),
  3. WWλ(v1,G)=1|Θ1|uΘ1i=1r|Θi|2(dλ(u,vi)+d2λ(u,vi)),
  4. H(v1,G)=1|Θ1|uΘ1i=1r|Θi|d(u,vi),
  5. H(v1,G,x)=1|Θ1|uΘ1i=1r|Θi|d(u,vi)xd(u,vi),
  6. Ht(v1,G)=1|Θ1|uΘ1i=1r|Θi|d(u,vi)+t,
  7. RCW(v1,G)=1|Θ1|uΘ1i=1r|Θi|D(G)+1d(u,vi).

Proof. Since PG is transitive, we get Wλ(v1,G)=1|Θ1|uΘ1zV(G)dλ(u,z)=1|Θ1|uΘ1i=1rzΘidλ(u,z)=1|Θ1|i=1rzΘiuΘ1dλ(u,z). For any i and any zΘi, there exist hiH satisfying hi(z)=vi. Therefore, zΘiuΘ1dλ(u,z)=zΘiuΘ1dλ(hi(u),hi(z))=zΘihi(u)Θ1dλ(hi(u),vi)=|Θi|hi(u)Θ1dλ(hi(u),vi)=|Θi|vΘ1dλ(v,vi). Consequently, we yield Wλ(v1,G)=1|Θ1|i=1ruΘ1zΘidλ(u,z)=1|Θ1|i=1r|Θi|vΘ1dλ(v,vi)=1|Θ1|vΘ1i=1r|Θi|dλ(v,vi). The remaining parts follows similarly, hence, we complete the proof.

In theorem 2.3, we can see that for a vertex v1Θ1, we do not need to compute all distances between v1 and V(G){v1}. It is enough to select one vertex from Θi. In real practice, we select a subgroup H so that Θ1 is as small as possible in order to simplify the calculation. Specially, if |Θ1|=1 (H fixes v1), we only count r1 times. Hence, we can give the following corollary.

Corollary 2.4. Let viΘi, i=1,2,,r. Assume that |Θ1|=1. We get

  1. Wλ(v1,G)=i=2r|Θi|dλ(v1,vi),
  2. WW(v1,G)=i=2r|Θi|d(v1,vi)+d2(v1,vi)2,
  3. WWλ(v1,G)=i=2r|Θi|dλ(v1,vi)+d2λ(v1,vi)2,
  4. H(v1,G)=i=2r|Θi|1d(v1,vi),
  5. H(v1,G,x)=i=2r|Θi|1d(v1,vi)xd(v1,vi),
  6. Ht(v1,G)=i=2r|Θi|1d(v1,vi)+t,
  7. RCW(v1,G)=i=2r|Θi|1D(G)+1d(v1,vi),
  8. π(v1,G)=i=2rd|Θi|(v1,vi),
  9. Π(v1,G)=ln2i=2rd|Θi|(v1,vi).

In order to reduce the computation steps of the distance-based topological indices, note that a large number of the molecular structures have the layered structure such that the different orbits have consecutive distances from a fixed vertex. In such a situation, we have following theorem.

Theorem 2.5. Assume that PG is transitive and HPG is a subgroup with orbits Θ1,Θ2,,Θr such that |Θi|=1. Let viΘi for i=1,2,,r. Then we have

  1. Wλ(G)=n2j=2r|Θkj|(j1)λ,
  2. WW(G)=n2j=2r|Θkj|j2j2,
  3. WWλ(G)=n2j=2r|Θkj|(j1)λ+(j1)2λ2,
  4. H(G)=n2j=2r|Θkj|1j1,
  5. H(G,x)=n2j=2r|Θkj|1j1xj1,
  6. Ht(G)=n2j=2r|Θkj|1j+t1,
  7. RCW(G)=n2j=2r|Θkj|1D(G)+2j,
  8. π(G)=(j=2r(j1)|Θkj|)n2,
  9. Π(G)=ln(j=2r(j1)|Θkj|)n2.

Proof. Science the molecular graphs are connected and all the elements in the same orbit have equal distances from v1, the orbits saturate the vacancy between Θ1 and Θk by means of their distances from v1. Since only r1 orbits different from Θ1 and d(v1,vk)r1, we infer that the orbits run consecutively between Θ1 and Θk, which reveals that the vertices v1,v2,,vr can be permuted into vk1,vk2,,vkr with k1=1 and kr=k satisfies d(v1,vkj)=j1, j=1,2,,r. Therefore, Wλ(G)=n2Wλ(v1,G)=n2i=2r|Θi|dλ(v1,vi)=n2j=2r|Θkj|dλ(v1,vkj)=n2j=2r|Θkj|(j1)λ. The remaining cases can be easily proved in similar fashion.

For the presentation of examples in the next section, we should mention that all the regular polyhedrons meet the conditions in the Theorem 2.5.

3. Computation Examples

In this section, we give five illustrative examples to explain our method. In the following contexts, we always assume that $n$ is the number of vertex in molecular graph $G$ and the regular polyhedrons meet the conditions of the theorem 2.5.

Example 3.1. [Computation on tetrahedron] The structure of tetrahedron (denoted by G1) can refer to Figure 1. Let PG1 be its point group. We need first determine the subgroup RPG1 of all the rotation in PG1. The elements consisting of R are as follows: (1) the identity; (2) rotations through the angle π about each of three axes joining the midpoints of opposite edges; (3) rotations through angles of 2π3 and 4π3 on the each of four axes joining vertices with centers of opposite faces. So, we have |R|=12.

Clearly, R and PG1 are transitive. Select H as the identity plus the set of all rotations around the axis joining v1 with the center of the opposite face through angles of 2π3 and 4π3 anticlockwise. We yield two orbits with representatives v1, v2 as presented in the Figure 1. Applying |Θi|=|P||Pv|, we infer |Θ1|=1, |Θ2|=3. According to theorem 2.5, we get

  1. Wλ(G1)=n2j=2r|Θkj|(j1)λ=423=6,
  2. WW(G1)=n2j=2r|Θkj|j2j2=42312(1+12)=6,
  3. WWλ(G1)=n2j=2r|Θkj|(j1)λ+(j1)2λ2=42312(1λ+12λ)=6,
  4. H(G1)=n2j=2r|Θkj|1j1=423=6,
  5. H(G1,x)=n2j=2r|Θkj|1j1xj1=423x=6x,
  6. Ht(G1)=n2j=2r|Θkj|1j+t1=42311+t=61+t,
  7. RCW(G1)=n2j=2r|Θkj|1D(G)+2j=42311+11=6,
  8. π(G1)=(j=2r(j1)|Θkj|)n2=1,
  9. Π(G1)=ln(j=2r(j1)|Θkj|)n2=ln2.

Example 3.2. [Computation on cube] \noindent The structure of cube (denoted as G2) can refer to Figure 2. In this case, the subgroup RPG2 of all the rotations consists of the follows: (1) rotations through the angle π on each of six axes joining midpoints of diagonally opposite edges; (2) rotations through angles of π2 and 3π2 about each of four axes joining extreme opposite vertices; (3) rotations through angles of π2,π, and 3π2 about each of three axes joining the centers of opposite faces. Thus, by simple computation, we get |R|=24.

Clearly, R and PG2 are both transitive. H is selected as in the first instance but the rotations are around the axis joining the two opposite vertices v1 and v3. We get four orbits with representatives v1, v2, v3 and v4 as presented in the Figure 2. In view of |Θi|=|P||Pv|, we yield |Θ1|=|Θ4|=1, |Θ2|=|Θ3|=3. Applying theorem 2.5, we get

  1. Wλ(G2)=n2j=2r|Θkj|(j1)λ=82(3+32λ+3λ)=12+122λ+43λ,
  2. WW(G2)=n2j=2r|Θkj|j2j2=82(32(1+1)+32(2+4)+3+92)=72,
  3. WWλ(G2)=n2j=2r|Θkj|(j1)λ+(j1)2λ2=82(32(1+1)+32(2λ+4λ)+3λ+9λ2)=12+6(2λ+4λ)+2(3λ+9λ),
  4. H(G2)=n2j=2r|Θkj|1j1=82(3+32+13)=583,
  5. H(G2,x)=n2j=2r|Θkj|1j1xj1=82(3x+32x2+13x3)=12x+6x2+43x3,
  6. Ht(G2)=n2j=2r|Θkj|1j+t1=4(31+t+32+t+13+t),
  7. RCW(G2)=n2j=2r|Θkj|1D(G)+2j=82(33+32+11)=14,
  8. π(G2)=(j=2r(j1)|Θkj|)n2=(13233)82=331776,
  9. Π(G2)=ln(j=2r(j1)|Θkj|)n2=ln5762.

Example 3.3. [Computation on octahedron] The structure of octahedron (denoted by G3) can refer to Figure 3. Obviously, we can get the octahedron by adding the midpoints of adjacent faces of the cube with edges. Form this point of view, its point group is the same as that of the cube.

Furthermore, PG3 is transitive, and all the rotations can be selected around the axis joining v1 and v3 which keep the octahedron invariant. Three orbits are obtained with representatives vi, i=1,2,3 as depicted in the Figure 3 and |Θ1|=|Θ3|=1 and |Θ2|=4. By theorem 2.5, we get
  1. Wλ(G3)=12+32λ,
  2. WW(G3)=21,
  3. WWλ(G3)=12+32(2λ+4λ),
  4. H(G3)=272,
  5. H(G3,x)=12x+3x2,
  6. Ht(G3)=121+t+32+t,
  7. RCW(G3)=9,
  8. π(G3)=8,
  9. Π(G3)=ln4.

Example 3.4. [Computation on icosahedron] The structure of icosahedron (denoted by G4) can refer to Figure 4. The rotation subgroup R of the point group consists: (1)the identity; (2) rotations through the angle π about each of fifteen axes joining midpoints of opposite edges; (3) rotations through angles of 2π3 and 4π3 about each of ten axes joining centers of opposite faces; (4) rotations through angles of 2π5,4π5,6π5, and 8π5 about each of six axes joining extreme opposite vertices. Therefore, we have |R|=60.

Figure 4. The structure of icosahedron G4
Furthermore, R and PG4 are transitive. H can be selected as the group generated by the 2π5-rotation around the axis joining v1 and v4. There are four orbits with representatives as shown in the Figure 4 and by |Θi|=|P||Pv| we get |Θ1|=|Θ4|=1, |Θ2|=|Θ3|=5. Applying theorem 2.5, we get
  1. Wλ(G4)=n2j=2r|Θkj|(j1)λ=122(5+52λ+3λ)=30+302λ+63λ,
  2. WW(G4)=n2j=2r|Θkj|j2j2=122(52(1+1)+52(2+4)+3+92)=156,
  3. WWλ(G4)=n2j=2r|Θkj|(j1)λ+(j1)2λ2=122(52(1+1)+52(2λ+4λ)+3λ+9λ2)=30+15(2λ+4λ)+3(3λ+9λ),
  4. H(G4)=n2j=2r|Θkj|1j1=122(5+52+13)=47,
  5. H(G4,x)=n2j=2r|Θkj|1j1xj1=122(5x+52x2+13x3)=30x+15x2+2x3,
  6. Ht(G4)=n2j=2r|Θkj|1j+t1=6(51+t+52+t+13+t),
  7. RCW(G4)=n2j=2r|Θkj|1D(G)+2j=122(53+52+11)=31,
  8. π(G4)=(j=2r(j1)|Θkj|)n2=(15253)122=966,
  9. Π(G4)=ln(j=2r(j1)|Θkj|)n2=ln9632.

Example 3.5. [Computation on dodecahedron] The structure of dodecahedron (denoted by G5) can refer to Figure 5. Similar as we discussed in the above examples, one can see that the dodecahedron and icosahedron have the same point group. Hence, H can be selected to be the group generated by the 2π3-rotation around the axis joining v1 and v6, and the reflection with respect to the plane containing v1,v2 and v6.

There are six orbits with representatives vi, i=1,2,,6 as shown in the Figure 5. By simple computation, we get |Θ1|=|Θ6|=1, |Θ2|=|Θ5|=3 and |Θ3|=|Θ4|=6. Thus, in terms of Theorem 2.5, we get
  1. Wλ(G5)=n2j=2r|Θkj|(j1)λ=202(3+62λ+63λ+34λ+5λ)=30+602λ+603λ+304λ+105λ,
  2. WW(G5)=n2j=2r|Θkj|j2j2=202(32(1+1)+3(2+4)+3(3+9)+32(4+16)+12(5+25))=1020,
  3. WWλ(G5)=n2j=2r|Θkj|(j1)λ+(j1)2λ2=202(32(1+1)+3(2λ+4λ)+3(3λ+9λ)+32(4λ+16λ)+12(5λ+25λ))=30+30(2λ+4λ)+30(3λ+9λ)+15(4λ+16λ)+5(5λ+25λ),
  4. H(G5)=n2j=2r|Θkj|1j1=202(3+62+63+34+15)=1792,
  5. H(G5,x)=n2j=2r|Θkj|1j1xj1=202(3x+62x2+63x3+34x4+15x5)=30x+30x2+20x3+152x4+2x5,
  6. Ht(G5)=n2j=2r|Θkj|1j+t1=202(31+t+62+t+63+t+34+t+15+t)=301+t+602+t+603+t+304+t+105+t,
  7. RCW(G5)=n2j=2r|Θkj|1D(G)+2j=202(35+64+63+32+11)=66,
  8. π(G5)=(j=2r(j1)|Θkj|)n2=(2636435)10,
  9. Π(G5)=ln(j=2r(j1)|Θkj|)n2=ln261360430510.

4. Conclusion

In this paper, we mainly report the approach on how to use group theory to determine the distance-based topological indices for certain important symmetry chemical structures. Since these Wiener related and other distance-based topological indices are widely applied in the analysis of both the boiling point and melting point of chemical compounds and QSPR/QSAR study, the promising prospects of their application for the chemical, medical and pharmacy engineering will be illustrated in the theoretical conclusion that is obtained in this article.

Acknowledgments

The authors thank the reviewers for their constructive comments in improving the quality of this paper.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

References

  1. Farahani, M. R., Jamil, M. K., & Imran, M. (2016). Vertex PIv topological index of titania carbon nanotubes TiO2(m,n). Appl. Math. Nonl. Sci, 1(1), 175-182. [Google Scholor]
  2. Jamil, M. K., Farahani, M. R., Imran, M., & Malik, M. A. (2016). Computing eccentric version of second zagreb index of polycyclic aromatic hydrocarbons PAHk. Appl. Math. Nonl. Sci, 1(1), 247-252.[Google Scholor]
  3. Gao, W., & Farahani, M. R. (2016). Computing the reverse eccentric connectivity index for certain family of nanocone and fullerene structures. Journal of Nanotechnology, Volume 2016, Article ID 3129561, 6 pages. [Google Scholor]
  4. Gao, W., Wang, W., & Farahani, M. R. (2016). Topological indices study of molecular structure in anticancer drugs. Journal of Chemistry, Volume 2016, Article ID 3216327, 8 pages.[Google Scholor]
  5. Gao, W., Farahani, M. R., & Shi, L. (2016). Forgotten topological index of some drug structures. Acta Medica Mediterranea, 32, 579-585. [Google Scholor]
  6. Gao, W., Shi, L., & Farahani, M. R. (2016). Distance-Based Indices for Some Families of Dendrimer Nanostars. Int. J. Appl. Math. 46(2) (2016), 168-186. [Google Scholor]
  7. Gao, W., & Wang, W. (2014). Second atom-bond connectivity index of special chemical molecular structures. Journal of Chemistry, Volume 2014, Article ID 906254, 8 pages. [Google Scholor]
  8. Gao, W., & Wang, W. (2015). The vertex version of weighted wiener number for bicyclic molecular structures. Computational and Mathematical Methods in Medicine, Volume 2015, Article ID 418106, 10 pages. [Google Scholor]
  9. Gao, W., & Wang, W. (2016). The eccentric connectivity polynomial of two classes of nanotubes. Chaos, Solitons & Fractals, 89, 290-294. [Google Scholor]
  10. Bondy, J. A., & Murty, U. S. R. (1976). Graph theory with applications (Vol. 290). London: Macmillan. [Google Scholor]
  11. Nikolić, S., Trinajstić, N., & Randić, M. (2001). Wiener index revisited. Chemical Physics Letters, 333(3-4), 319-321. [Google Scholor]
  12. Vukičević, D., & Žerovnik, J. (2003). New indices based on the modified Wiener indices. MATCH: communications in mathematical and in computer chemistry, 47, 119-132.[Google Scholor]
  13. Vukičević, D., & Gutman, I. (2003). Note on a class of modified Wiener indices. MATCH: communications in mathematical and in computer chemistry, 47, 107-117.
  14. Lim, T. C. (2004). Mass-modified Wiener indices and the boiling points for lower chloroalkanes. Acta Chim. Slov, 51(4), 611-618. [Google Scholor]
  15. Gorse, M., & Zerovnik, J. (2004). A remark on modified Wiener indices. MATCH-Communications in Mathematical and in Computer Chemistry, (50), 109-116.[Google Scholor]
  16. Vukičević, D., & Graovac, A. (2004). On modified Wiener indices of thorn graphs. MATCH Communications in mathematical and in computer chemistry, 50(50), 93-108. [Google Scholor]
  17. Gutman, I., Vukičević, D., & Žerovnik, J. (2004). A class of modified Wiener indices. Croatica chemica acta, 77(1-2), 103-109. [Google Scholor]
  18. Gutman, I. (2003). Hyper-Wiener index and Laplacian spectrum. Journal of the Serbian Chemical Society, 68(12), 949-952. [Google Scholor]
  19. Gutman, I., & Furtula, B. (2003). Hyper-Wiener index vs. Wiener index. Two highly correlated structure-descriptors. Monatshefte für Chemie/Chemical Monthly, 134(7), 975-981.[Google Scholor]
  20. Eliasi, M., & Taeri, B. (2007). Hyper-Wiener index of zigzag polyhex nanotorus. Ars Combinatoria, 85, 307-318.[Google Scholor]
  21. Eliasi, M., & Taeri, B. (2008). Hyper-Wiener index of zigzag polyhex nanotubes. The ANZIAM Journal, 50(1), 75-86. [Google Scholor]
  22. Iranmanesh, A., Alizadeh, Y., & Mirzaie, S. (2009). Computing Wiener polynomial, Wiener index and Hyper Wiener index of C80 fullerene by GAP program. Fullerenes, Nanotubes and Carbon Nanostructures, 17(5), 560-566. [Google Scholor]
  23. Yazdani, J., & Bahrami, A. (2009). Hyper-Wiener index of symmetric y-junction nanotubes. Digest Journal of Nanomaterials & Biostructures (DJNB), 4(3), 479-481. [Google Scholor]
  24. Behtoei, A., Jannesari, M., & Taeri, B. (2009). Maximum Zagreb index, minimum hyper-Wiener index and graph connectivity. Applied Mathematics Letters, 22(10), 1571-1576. [Google Scholor]
  25. Mansour, T., & Schork, M. (2010). Wiener, hyper-Wiener, detour and hyper-detour indices of bridge and chain graphs. Journal of mathematical chemistry, 47(1), 72. [Google Scholor]
  26. Heydari, A. (2010). Hyper Wiener Index of C4C8(S) Nanotubes. Current Nanoscience, 6(2), 137-140. [Google Scholor]
  27. Ashrafi, A. R., Hamzeh, A., & Hossein-Zadeh, S. (2011). Computing Zagreb, Hyper-Wiener and Degree-Distance indices of four new sums of graphs. Carpathian Journal of Mathematics, 27, 153-164.[Google Scholor]
  28. Heydari, A. (2012). On the hyper Wiener index of hexagonal chains. Optoelectronics and Advanced Materials-Rapid Communications, 6(3-4), 491-494.
  29. Plavšić, D., Nikolić, S., Trinajstić, N., & Mihalić, Z. (1993). On the Harary index for the characterization of chemical graphs. Journal of Mathematical Chemistry, 12(1), 235-250. [Google Scholor]
  30. Ivanciuc, O., Balaban, T. S., & Balaban, A. T. (1993). Design of topological indices. Part 4. Reciprocal distance matrix, related local vertex invariants and topological indices. Journal of Mathematical Chemistry, 12(1), 309-318. [Google Scholor]
  31. Das, K. C., Xu, K., Cangul, I. N., Cevik, A. S., & Graovac, A. (2013). On the Harary index of graph operations. Journal of Inequalities and Applications, 2013(1), 339. [Google Scholor]
  32. Zhou, B., Cai, X., & Trinajstić, N. (2009). On reciprocal complementary Wiener number. Discrete applied mathematics, 157(7), 1628-1633. [Google Scholor]
  33. Gutman, I., Linert, W., Lukovits, I., & Tomović, Ž. (2000). The multiplicative version of the Wiener index. Journal of chemical information and computer sciences, 40(1), 113-116. [Google Scholor]
  34. Gutman, I., Linert, W., Lukovits, I., & Tomović, Ž. (2000). On the multiplicative Wiener index and its possible chemical applications. Monatshefte für Chemie/Chemical Monthly, 131(5), 421-427. [Google Scholor]