Some topological indices of the family of nanostructures of polycyclic aromatic hydrocarbons (PAHs)

Molecular descriptors play a significant role in many areas such as chemistry and pharmacology. Among these, topological indices have a prominent place. In this paper, we focus on the structure of polycyclic aromatic hydrocarbons and calculate theta, Pi and Sadhana indices of nanostructures. These indices on the ground of quasi-orthogonal cut ‘qoc’ edge strips in molecular graph. Also, the exact formulae for connectivity indices of nanotubes and nanotori are presented in this manuscript. These topological indices are useful for surveying structure of nanostructures, which have a relationship with degrees of their vertices.


INTRODUCTION
Graph theory models have extensively been used as predictors of properties of chemical compounds scientist who considered graph theoretical problems of nanotechnology, and some works on computing topological indices of nanostructure, nanotubes and nanotori are presented in many papers (Diudea & John, 2001;Diudea & Kirby, 2001;Diudea, 2002;Diudea et al., 2003;2004).
The present study is a continuation of our previous studies Soleimani et al., 2015; 2016) on computing some topological indices of nanostructures.

METHODOLOGY
From the point of graph theory, all organic molecular structures can be drawn as graphs in which atoms and bonds are represented by vertices and edges, respectively. Let be a simple graph with vertex set and edge set . The edge connecting the vertices and will be denoted by . The degree of the vertex , denoted by in the underlying graph. The distance between and in , , is the length of a shortest path in . Two edges e = uv and of are called co-distant, e co The above relation co any edge e of but in general is not transitive. A graph is called a co-graph if the relation co is also transitive and thus an equivalence relation.

Let
be the set of edges in that are co-distant to . The set can be obtained by an orthogonal edge cutting procedure: take a straight line segment, orthogonal to the edge e, and intersect it and all other edges (of a polygonal plane graph) parallel to e. The set of these intersections is called an orthogonal cut (oc for short) of , with respect to e. If is a co-graph then its orthogonal cuts form a partition of : , ,

for and
If any two consecutive edges e and of a plane graph of an edge-cut sequence are topologically parallel within the same face of the covering, such a sequence is called a quasi-orthogonal cut (qoc) strip. Obviously, any orthogonal cut strip is a qoc strip but the reverse is not always true. This means the transitivity relation of the co relation is not necessarily obeyed. Topological index is a real number that is derived from molecular graphs of chemical compounds. There are several topological

Connectivity indices
physical properties of alkanes to the degree of branching across an isomeric series. The degree of branching of a connectivity index . Kier and Hall (1986) extended this for heteroatoms.
Let be a simple connected graph of order n. For an integer , the m-order connectivity index of an organic molecule whose molecule graph ... (4) where (for simplicity) runs over all paths of length in , and denote the degree of vertex .
index, named the sum-connectivity index. It has been found that the sum-connectivity index correlates well and it is frequently applied in quantitative structure property and structure-activity studies. The m-order sum-connectivity index of ... (5) follows: ... (6) The second follows: ...(7)

RESULTS AND DISCUSSION
The explicit formulae for the theta, Pi, Sadhana, secondorder connectivity and second-order sum-connectivity indices of vertical pentacenic nanotube, horizontal pentacenic nanotube and pentacenic nanotori are presented in this section.
Polycyclic aromatic hydrocarbons (PAHs) are organic compounds containing only carbon and hydrogen that are composed of multiple aromatic rings. Pentacenes linearly-fused benzene rings (Soleimani et al., 2014).

Remark 1:
We denote a 2-dimensional lattice of vertical pentacenic nanotube by ( Figure 1) for every e in . By using the cut method and Figure 1, one can see that there are some distinct cases of qoc strips. We denote the corresponding edges by . Regarding the different possible cases which and can be chosen, the following cases are considered.

Case 1.
: Type of edges Number of Number of co-distant edges qoc  If using the data in Table 1, the theta index of , for , can be written as

March 2018
Journal of the National Science Foundation of Sri Lanka 46 (1) Sadhana indices, the proof is straightforward.

Theorem 3 The second order connectivity index of is as follows:
Proof. as a number of 2-edges paths with vertices of degree i, j and k, respectively. It is obvious, = to be the number of edges connecting the three vertices of degree 2, 3 and 2 (the red path in Figure 2), to be the number of edges connecting the three vertices of degree 3, 2 and 3 (the blue path in Figure 2), to be the number of edges connecting three vertices of degree 2, 3 and 3 (the green path in Figure 2), and to be the number of edges connecting the three vertices of degree 3 (the pink path in Figure 2).    Table 3, we have the following computations: If using the data in Table 2, similar to the pervious case, we have Therefore, Summing up contributions of the two parts completes the proof.
Similar to the proof of Theorem 1, we can prove the following theorems: Proof. ,

Remark 2:
We denote a 2-dimensional lattice of horizontal pentacenic nanotube by ( Figure 3). Now we consider the molecular graph . It is easy to see that and . Notice that the edges in the top gain a tube in this way.

Example 2:
Consider the graph of 2-dimensional lattice of nanotube as depicted in Figure 3.
Since the edge number of is equal to 388, thus we obtain Also, In general, for , we have the following theorems.

Theorem 5
The theta, Pi and Sadhana indices of , are computed as: Proof. Let be the graph of -pentacenic nanotube. Tables 4 and 5 show the number of codistant edges in . By using these tables the proof is straightforward.   (3,3,3) 66pq -20q Table 6: Categorisation of all 2-edges paths Now, by using the results in Table 6, we have the following computations: The second-order sum-connectivity index of is as follows: Proof. . Proof. d 233 to be the number of edges connecting three vertices of degree 2, 3 and 3 (the green path in Figure 4), d 322 to be the number of edges connecting the three vertices of degree 3, 2 and 2 (the orange path in Figure 4), and d 333 to be the number of edges connecting the three vertices of degree 3 (the pink path in Figure 4). Remark 3: We denote a 2-dimensional lattice of pentacenic nanotori by ( Figure 5). Now we consider the molecular graph. It is easy to see that and . Notice that the this way.   Proof. By using Table 7 and Table 8, we give explicit computing formulae for topological indices of nanotori, as shown in Figure 5.

Theorem 9
The second-order connectivity and secondorder sum-connectivity indices of are computed as follows:

Proof.
d 333 to be the number of edges connecting three vertices of degree 3. It is easy to see that the number of 2-edges paths in is equal to 66pq. Therefore, the proof is clear.

CONCLUSION
In this paper, a simple method enabling to compute the number of co-distant edges and number of qoc of vertical and horizontal pentacenic nanotube and nanotori is presented. By our calculation it is possible to evaluate the omega and its related counting polynomials of these nanostructures for future works.