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.
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 \(f:G\rightarrow \mathcal{R}^{+}\). 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, $$W(G)=\sum_{\{u,v\}\subseteq V(G)}d(u,v).$$ The modified Wiener index was introduced by Nikolić et al. [11] as the extension of the Wiener index which was defined as $$W_{\lambda}(G)=\sum_{\{u,v\}\subseteq V(G)}d^{\lambda}(u,v).$$ 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 \(\lambda\)-modified hyper-Wiener index are defined as $$WW(G)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{2}(d(u,v)+d^{2}(u,v))$$ and $$WW_{\lambda}(G)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{2}(d^{\lambda}(u,v)+d^{2\lambda}(u,v)),$$ 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 $$H(G)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{d(u,v)}.$$ Its corresponding Harary polynomial can be defined as $$H(G,x)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{d(u,v)}x^{d(u,v)}.$$ The second and third Harary indices are defined as $$H_{1}(G)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{d(u,v)+1},$$ $$H_{2}(G)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{d(u,v)+2}.$$ More generally, the generalized Harary index was introduced by Das et al. [31] which is defined as $$H_{t}(G)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{d(u,v)+t},$$ where \(t\in\mathcal{N}\) 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 $$RCW(G)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{D(G)+1-d(u,v)},$$ 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 $$\pi(G)=\prod_{\{u,v\}\subseteq V(G)}d(u,v).$$ The logarithm of multiplicative Wiener index was defined as $$\Pi(G)=\ln\sqrt{2\prod_{\{u,v\}\subseteq V(G)}d(u,v)}.$$
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.
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 \(p\in P\) that satisfies \(p(v)=u\) for two vertices \(v\) and \(u\), then we define an equivalence binary relation denoted by \(v\sim u\). By means of this equivalent relation, the vertex set is divided into several equivalence classes: \(\Theta_{1},\cdots,\Theta_{r}\). Each \(\Theta_{i}\) can be called an orbit of point group \(P\), and the number of vertices of each orbit \(|\Theta_{i}|\) is called the length of the orbit \(\Theta_{i}\). By the knowledge of group theory, we know that if \(v\in \Theta_{i}\) then \(|\Theta_{i}|=\frac{|P|}{|P_{v}|}\), where \(P_{v}=\{p|p\in P,p(v)=v\}\). 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.
SetTheroem 2.1. Let \(H\unlhd P_{G}\) be a subgroup of \(P_{G}\), and \(\Theta_{1}, \Theta_{2},\cdots, \Theta_{r}\) are the orbits of \(H\) and \(u_{i}\in\Theta_{i}\), \(i=1,2,\cdots,r\). Then, we have
Proof. We only prove for \(W_{\lambda}(G)\). The remaining cases can be proved in similar fashion. Since \(W_{\lambda}(u,G)\) is equal to the sum of all vertices in the same orbit, we infer \begin{eqnarray*} \sum_{w\in V(G)}W_{\lambda}(w,G)&=&\sum_{i=1}^{r}\sum_{w\in \Theta_{i}}W_{\lambda}(w,G)\\&=&\sum_{i=1}^{r}|\Theta_{i}|W_{\lambda}(u_{i},G)\\ &=&\sum_{i=1}^{r}|\Theta_{i}|\sum_{j=1}^{r}\sum_{y\in \Theta_{i}}d^{\lambda}(u_{i},y)\\&=&\sum_{i=1}^{r}\sum_{j=1}^{r}|\Theta_{i}|\sum_{y\in \Theta_{i}}d^{\lambda}(u_{i},y)\\ &=&2\sum_{i=1}^{r}\sum_{j=i+1}^{r}|\Theta_{i}|\sum_{y\in\Theta_{i}}d^{\lambda}(y,u_{i})+\sum_{i=1}^{r}|\Theta_{i}|\sum_{z\in \Theta_{i}}d^{\lambda}(z,u_{i}). \end{eqnarray*} Hence, in terms of (1), \begin{eqnarray*}W_{\lambda}(G)&=&\frac{1}{2}\sum_{w\in V(G)}W_{\lambda}(w,G)\\ &=&\sum_{i=1}^{r}\sum_{j=i+1}^{r}|\Theta_{i}|\sum_{u\in\Theta_{j}}d^{\lambda}(u,u_{i})+\frac{1}{2}\sum_{i=1}^{r}|\Theta_{i}|\sum_{v\in \Theta_{i}}d^{\lambda}(v,u_{i}). \end{eqnarray*} 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 \(P_{G}\) of the molecular graph is transitive. Then for any \(v\in V(G)\), we have
Theorem 2.3. Let \(v_{i}\in \Theta_{i}\), \(i=1,2,\cdots,r\). Then,
Proof. Since \(P_{G}\) is transitive, we get \begin{eqnarray*} W_{\lambda}(v_{1},G)&=&\frac{1}{|\Theta_{1}|}\sum_{u\in \Theta_{1}}\sum_{z\in V(G)}d^{\lambda}(u,z)\\ &=&\frac{1}{|\Theta_{1}|}\sum_{u\in \Theta_{1}}\sum_{i=1}^{r}\sum_{z\in \Theta_{i}}d^{\lambda}(u,z)\\ &=&\frac{1}{|\Theta_{1}|}\sum_{i=1}^{r}\sum_{z\in \Theta_{i}}\sum_{u\in \Theta_{1}}d^{\lambda}(u,z). \end{eqnarray*} For any \(i\) and any \(z\in \Theta_{i}\), there exist \(h_{i}\in H\) satisfying \(h_{i}(z)=v_{i}\). Therefore, \begin{eqnarray*} \sum_{z\in \Theta_{i}}\sum_{u\in \Theta_{1}}d^{\lambda}(u,z)&=&\sum_{z\in \Theta_{i}}\sum_{u\in \Theta_{1}}d^{\lambda}(h_{i}(u),h_{i}(z))\\ &=&\sum_{z\in \Theta_{i}}\sum_{h_{i}(u)\in \Theta_{1}}d^{\lambda}(h_{i}(u),v_{i})\\&=&|\Theta_{i}|\sum_{h_{i}(u)\in \Theta_{1}}d^{\lambda}(h_{i}(u),v_{i})\\ &=&|\Theta_{i}|\sum_{v\in\Theta_{1}}d^{\lambda}(v,v_{i}). \end{eqnarray*} Consequently, we yield \begin{eqnarray*} W_{\lambda}(v_{1},G)&=&\frac{1}{|\Theta_{1}|}\sum_{i=1}^{r}\sum_{u\in \Theta_{1}}\sum_{z\in \Theta_{i}}d^{\lambda}(u,z)\\ &=&\frac{1}{|\Theta_{1}|}\sum_{i=1}^{r}|\Theta_{i}|\sum_{v\in\Theta_{1}}d^{\lambda}(v,v_{i})\\ &=&\frac{1}{|\Theta_{1}|}\sum_{v\in\Theta_{1}}\sum_{i=1}^{r}|\Theta_{i}|d^{\lambda}(v,v_{i}). \end{eqnarray*} The remaining parts follows similarly, hence, we complete the proof.
In theorem 2.3, we can see that for a vertex \(v_{1}\in\Theta_{1}\), we do not need to compute all distances between \(v_{1}\) and \(V(G)-\{v_{1}\}\). It is enough to select one vertex from \(\Theta_{i}\). In real practice, we select a subgroup \(H\) so that \(\Theta_{1}\) is as small as possible in order to simplify the calculation. Specially, if \(|\Theta_{1}|=1\) (\(H\) fixes \(v_{1}\)), we only count \(r-1\) times. Hence, we can give the following corollary.Corollary 2.4. Let \(v_{i}\in\Theta_{i}\), \(i=1,2,\cdots,r\). Assume that \(|\Theta_{1}|=1\). We get
Theorem 2.5. Assume that \(P_{G}\) is transitive and \(H\unlhd P_{G}\) is a subgroup with orbits \(\Theta_{1},\Theta_{2},\cdots,\Theta_{r}\) such that \(|\Theta_{i}|=1\). Let \(v_{i}\in\Theta_{i}\) for \(i=1,2,\cdots,r\). Then we have
Proof. Science the molecular graphs are connected and all the elements in the same orbit have equal distances from \(v_{1}\), the orbits saturate the vacancy between \(\Theta_{1}\) and \(\Theta_{k}\) by means of their distances from \(v_{1}\). Since only \(r-1\) orbits different from \(\Theta_{1}\) and \(d(v_{1},v_{k})\ge r-1\), we infer that the orbits run consecutively between \(\Theta_{1}\) and \(\Theta_{k}\), which reveals that the vertices \(v_{1},v_{2},\cdots,v_{r}\) can be permuted into \(v_{k_{1}},v_{k_{2}},\cdots,v_{k_{r}}\) with \(k_{1}=1\) and \(k_{r}=k\) satisfies \(d(v_{1},v_{k_{j}})=j-1\), \(j=1,2,\cdots,r\). Therefore, \begin{eqnarray*} W_{\lambda}(G)&=&\frac{n}{2}W_{\lambda}(v_{1},G)\\&=&\frac{n}{2}\sum_{i=2}^{r}|\Theta_{i}|d^{\lambda}(v_{1},v_{i})\\ &=&\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|d^{\lambda}(v_{1},v_{k_{j}})\\&=&\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|(j-1)^{\lambda}. \end{eqnarray*} 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.Example 3.1. [Computation on tetrahedron] The structure of tetrahedron (denoted by \(G_{1}\)) can refer to Figure 1. Let \(P_{G_{1}}\) be its point group. We need first determine the subgroup \(R\unlhd P_{G_{1}}\) of all the rotation in \(P_{G_{1}}\). The elements consisting of \(R\) are as follows: (1) the identity; (2) rotations through the angle \(\pi\) about each of three axes joining the midpoints of opposite edges; (3) rotations through angles of \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\) on the each of four axes joining vertices with centers of opposite faces. So, we have \(|R|=12\).
Clearly, \(R\) and \(P_{G_{1}}\) are transitive. Select \(H\) as the identity plus the set of all rotations around the axis joining \(v_{1}\) with the center of the opposite face through angles of \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\) anticlockwise. We yield two orbits with representatives \(v_{1}\), \(v_{2}\) as presented in the Figure 1. Applying \(|\Theta_{i}|=\frac{|P|}{|P_{v}|}\), we infer \(|\Theta_{1}|=1\), \(|\Theta_{2}|=3\). According to theorem 2.5, we get
Example 3.2. [Computation on cube] \noindent The structure of cube (denoted as \(G_{2}\)) can refer to Figure 2. In this case, the subgroup \(R\unlhd P_{G_{2}}\) of all the rotations consists of the follows: (1) rotations through the angle \(\pi\) on each of six axes joining midpoints of diagonally opposite edges; (2) rotations through angles of \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\) about each of four axes joining extreme opposite vertices; (3) rotations through angles of \(\frac{\pi}{2},\pi\), and \(\frac{3\pi}{2}\) about each of three axes joining the centers of opposite faces. Thus, by simple computation, we get \(|R|=24\).
Clearly, \(R\) and \(P_{G_{2}}\) are both transitive. \(H\) is selected as in the first instance but the rotations are around the axis joining the two opposite vertices \(v_{1}\) and \(v_{3}\). We get four orbits with representatives \(v_{1}\), \(v_{2}\), \(v_{3}\) and \(v_{4}\) as presented in the Figure 2. In view of \(|\Theta_{i}|=\frac{|P|}{|P_{v}|}\), we yield \(|\Theta_{1}|=|\Theta_{4}|=1\), \(|\Theta_{2}|=|\Theta_{3}|=3\). Applying theorem 2.5, we get
Example 3.3. [Computation on octahedron] The structure of octahedron (denoted by \(G_{3}\)) 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.
Example 3.4. [Computation on icosahedron] The structure of icosahedron (denoted by \(G_{4}\)) can refer to Figure 4. The rotation subgroup \(R\) of the point group consists: (1)the identity; (2) rotations through the angle \(\pi\) about each of fifteen axes joining midpoints of opposite edges; (3) rotations through angles of \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\) about each of ten axes joining centers of opposite faces; (4) rotations through angles of \(\frac{2\pi}{5},\frac{4\pi}{5},\frac{6\pi}{5}\), and \(\frac{8\pi}{5}\) about each of six axes joining extreme opposite vertices. Therefore, we have \(|R|=60\).
Example 3.5. [Computation on dodecahedron] The structure of dodecahedron (denoted by \(G_{5}\)) 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 \(\frac{2\pi}{3}\)-rotation around the axis joining \(v_{1}\) and \(v_{6}\), and the reflection with respect to the plane containing \(v_{1},v_{2}\) and \(v_{6}\).
