Journal of Chemistry
Volume 2014 (2014), Article ID 906254, 8 pages
http://dx.doi.org/10.1155/2014/906254

Research Article

Second Atom-Bond Connectivity Index of Special Chemical Molecular Structures

Wei Gao¹ and Weifan Wang²

¹School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China
²Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received 7 July 2014; Accepted 27 August 2014; Published 15 October 2014

Academic Editor: Demeter Tzeli

Copyright © 2014 Wei Gao and Weifan Wang. 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.

In theoretical chemistry, the second atom-bond connectivity index was introduced to measure the stability of alkanes and the strain energy of cycloalkanes. In this paper, we determine the second atom-bond connectivity index of unilateral polyomino chain and unilateral hexagonal chain. Furthermore, the second ABC indices of V-phenylenic nanotubes and nanotori are presented.

1. Introduction

One of the most important applications of chemical graph theory is to measure chemical, physical, and pharmaceutical properties of molecules called alkanes. Several indices that relied on the graphical structure of the alkanes are defined and employed to model both the melting point and boiling point of the molecules. Molecular graph is a topological representation of a molecule such that each vertex represents an atom of molecule, and covalent bonds between atoms are represented by edges between the corresponding vertices.

Specifically, topological index can be regarded as a score function , with this property that if two molecular graphs and are isomorphic. There are several vertex distance-based and degree-based indices which are introduced to analyze the chemical properties of molecule graph, for instance, Wiener index, PI index, Szeged index, and atom-bond connectivity index. Several papers contributed to determining the indices of special molecular graphs (see Yan et al. [1, 2], Gao and Shi [3], Xi and Gao [4], and Dou et al. [5] for more detail).

All (molecular) graphs considered in this paper are finite, loopless, and without multiple edges. Let be a (molecular) graph with vertex set and edge set . All graph notations and terminologies used but undefined in this paper can be found in [6].

The atom-bond connectivity index (shortly, index) is defined by Estrada et al. [7] as

Das et al. [8] characterized the molecular graphs as extremal with regard to index. Furtula et al. [9] found the chemical trees with extremal index. Vassilev and Huntington [10] identified certain classes of edges that are important and occur frequently in chemical trees and continuously studied the chemical trees with extremal index in terms of learning how the removal of a certain edge takes place. Chen et al. [11] obtained the atom-bond connectivity index of the zig-zag chain polyomino molecular graphs. Furthermore, they determined the sharp upper bound on the atom-bond connectivity index of catacon densed polyomino molecular graphs with squares and determined the corresponding extremal molecular graphs.

Recently, Graovac and Ghorbani [12] defined a new version of the atom-bond connectivity index, that is, the second atom-bond connectivity index (shortly, second index):

where, for each , is the number of vertices closer to vertex than vertex and defines similarly. In the chemistry applications, second index is used to model both the boiling point and melting point of the molecules. Hence, it is also applied to the pharmaceutical field. Rostami et al. [13] calculated some upper bounds for the second atom-bond connectivity index.

Although there have been several advances in atom-bond connectivity index of molecular graphs, the study of second atom-bond connectivity index of special chemical structures has been largely limited. In addition, as widespread and critical chemical structures, polyomino system, hexagonal system, V-phenylenic nanotubes, and nanotori are widely used in medical science and pharmaceutical field. As an example, polyomino chain is one of the basic chemical structures and exists widely in benzene and alkali molecular structures. For these reasons, we have attracted tremendous academic and industrial interests to research the second atom-bond connectivity of these molecular structures from a mathematical point of view.

The contribution of our paper is threefold. First, we compute the second atom-bond connectivity index of unilateral polyomino chain. Then, the second atom-bond connectivity index of unilateral hexagonal chain is calculated. At last, we derive the second atom-bond connectivity index of V-phenylenic nanotubes and nanotori.

2. Second Atom-Bond Connectivity Index of Polyomino Chain

From the view of graph theory, polymino is a finite 2-connected planar graph and each interior face is surrounded by a square with length 4. Polyomino chain is one class of polyomino such that the connection of centres for adjacent squares constitutes a path , where is the centre of th square. Polyomino chain is called a linear chain if the subgraph induced by all 3-degree vertices is a graph with squares. Furthermore, polyomino chain is called a zig-zag chain if the subgraph induced by all vertices with degree is path with edges. In what follows, we use and to denote linear polyomino chain and zig-zag polyomino chain, respectively. For the structure of and , refer to Figure 1.

Figure 1: The structure of and .

Use the similar technology raised in Gutman and Klavžar [14], and we define elementary cut as follows. Choose an edge of the polyomino system and draw a straight line through the center of , orthogonal on . This line will intersect the perimeter in two end points, and . The straight line segment whose end points are and is the elementary cut, intersecting the edge . A fragment in polyomino chain is just maximal linear chain which includes the squares in start and end vertices. Let be the length of fragment which denotes the number of squares it contained. Let be a polyomino chain with squares consisting of fragment sequence   . Denote   . It is not difficult to verify that and , . For the th fragment of polyomino chain, the cut of this fragment is the cut which intersects with parallel edges of squares in this fragment. A fragment is called horizontal fragment if its cut parallels the horizontal direction and called vertical fragment if its cut parallels the vertical direction. Unilateral polyomino chain is a special kind of polyomino chain such that, for each vertical fragment, two horizontal fragments (if exists) are adjacent and it appears in the left and right sides, respectively.

Our main result in this section stated as follows presents the second atom-bond connectivity index of unilateral polyomino chain.

Theorem 1. Let be a unilateral polyomino chain consisting of fragment   , and let    be the length of each fragment. Then, one has

Proof. The cuts in are divided into two types: I-type and II-type (see Figure 2). An edge is called I-type (or II-type) if it intersects with I-type (or II-type) cut. Now, we consider the following two cases.
Case 1. If edge is I-type in th square of th fragment (i.e., is edge which is passed by dotted line in Figure 2), we observe that there is such edges in th fragment.
Subcase 1.1.  If , then we have

Subcase 1.2. If , then we obtain

Subcase 1.3. If , then we get

Case 2. If edge is II-type in th square of th fragment. (i.e., is the edge which is passed by real line in Figure 2).
Subcase 2.1. If , then we yield

Subcase 2.2. If , then we infer

Subcase 2.3. If , then we deduce

Hence, the desired conclusion is obtained by combining the above case and the definition of the second atom-bond connectivity index.

Figure 2: I-type cut and II-type cut of unilateral polyomino chain.

Corollary 2. Let be the linear chain with squares. Then, one has

Proof. By the definition of linear chain, we have ,  ,  . In terms of Theorem 1, we immediately get the result.

Corollary 3. Let be the zig-zag chain with squares. Then, one has

Proof. By the virtue of the definition of the zig-zag chain, we have and . In view of Theorem 1, the result is immediately obtained.

3. Second Atom-Bond Connectivity Index of Hexagonal Chain

Hexagonal chain is one class of hexagonal system consisting of hexagonal. In hexagonal chain, each of two hexagonal has one common edge or no common vertex. Two hexagonal are adjacent if they have common edge. No three or more hexagonal share one vertex. Each hexagonal has two adjacent hexagonal except hexagonal in terminus, and each hexagonal chain has two hexagonal in terminus.

It is easy to verify that the hexagonal chain with   hexagonal has vertices and edges. Let and be the linear hexagonal chain and zig-zag hexagonal chain, respectively. For the chemical structure of and , one can refer to Figure 3 for more detail.

Figure 3: The structure of and .

Again, we use the similar trick which was presented in Gutman and Klavžar [14], and we define elementary cut as follows. Choose an edge of the hexagonal system and draw a straight line through the center of , orthogonal on . This line will intersect the perimeter in two end points and . The straight line segment whose end points are and is the elementary cut, intersecting the edge . A fragment in hexagonal chain is just maximal linear chain which includes the hexagonal in start and end vertices. Let be the length of fragment which denotes the number of hexagonal contained. Let be a hexagonal chain with   hexagonal consisting of fragment sequence   . Denote   . Then, we verify that since each two adjacent fragments have one common hexagonal. For the th fragment of hexagonal chain, the cut of this fragment is the cut which intersects with parallel edges of hexagonal in this fragment. A fragment is called horizontal fragment if its cut parallels the horizontal direction; otherwise, it is called inclined fragment. Unilateral hexagonal chain is a special class of hexagonal chain such that the cut for each inclined fragment at the same angle with a horizontal direction. As an example, Figure 4 shows a structure of unilateral hexagonal chain. Clearly, linear hexagonal chain is a unilateral hexagonal chain with one fragment, and zig-zag is a unilateral hexagonal chain with fragments.

Figure 4: I-type cut and II-type cut of unilateral hexagonal chain.

We now show the second atom-bond connectivity index of unilateral hexagonal chain.

Theorem 4. Let be a unilateral hexagonal chain consisting of fragment   , and let    be the length of each fragment. Then, one has

Proof. The cuts in are divided into two types: I-type and II-type (see Figure 4). An edge is called I-type if it intersects with I-type cut. Also, an edge is called II-type if it intersects with II-type cut. In what follows, we consider two situations according to the type of an edge.
Case 1. If edge is I-type in th square of th fragment (i.e., is edge which is passed by dotted line in Figure 4).
Subcase 1.1. If , then we have

Subcase 1.2. If , then, we obtain

Subcase 1.3. If , then we get

Case 2. If edge is II-type in th square of th fragment (i.e., is the edge which is passed by real line in Figure 4).
Subcase 2.1. If , then we yield

Subcase 2.2. If , then we infer

Subcase 2.3. If , then we deduce

In particular, if in Subcase 2.3, we have and .
Hence, we get the desired result by combining the above cases and the definition of the second atom-bond connectivity index.

Corollary 5. Let be the linear chain with   hexagonal. Then, one has

Proof. By the definition of linear chain, we check that ,  ,  . In terms of Theorem 4, we shortly get the result.

Corollary 6. Let be the zig-zag chain with hexagonal. Then, one has

Proof. Using the definition of zig-zag chain, we verify that and . In view of Theorem 4, the corollary is immediately yielded.

4. Second Atom-Bond Connectivity Index of V-Phenylenic Nanotubes and Nanotori

The notations in this section follow from Diudea [15]. In what follows, we use and to express and for short, respectively. The molecular structures V-phenylenic nanotube and V-phenylenic nanotorus are denoted by and , respectively.

The next result shows the representation of , and the structure of is described in Figure 5.

Figure 5: The structure of V-phenylenic nanotube.

Theorem 7. Consider

where and are presented in the process of the proof.

Proof. First, we note that . To compute the second atom-bond connectivity index of , we assume that , , and are the set of all vertical, oblique, and horizontal edges, respectively. Then, we infer

Now, we calculate by considering the following two cases.
Case 1 (). Under this situation, we deduce

where and . Hence, in this case, we infer that the second atom-bond connectivity index of is

Case 2 (). In this case, we get

Thus, the corresponding second atom-bond connectivity index is

The result is archived.

Our last result computes the second atom-bond connectivity index of and the structure of presented in Figure 6.

Figure 6: The structure of V-phenylenic nanotorus.

Theorem 8. Consider

where and are presented in the process of the proof.

Proof. In order to prove the theorem, we use a similar trick as in Theorem 7. Obviously, . We assume that , , and are the set of all vertical, oblique, and horizontal edges, respectively. Then we deduce

Now, we calculate according to the relationship between and .
Case 1 (). Under this assumption, we infer

where and . Hence, the second atom-bond connectivity index of for is

Case 2 (). In this case, we have

Thus, the corresponding second atom-bond connectivity index is

The proof is completed.

Conflict of Interests

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

Acknowledgments

The authors thank the reviewers for their constructive comments in improving the quality of this paper. They also would like to thank the anonymous referees for providing them with constructive comments and suggestions. The research is partially supported by NSFC (nos. 11401519, 11371328, and 11471293).

References

L. Yan, Y. Li, W. Gao, and J. Li, “PI index for some special graphs,” Journal of Chemical and Pharmaceutical Research, vol. 5, no. 11, pp. 260–264, 2013.

L. Yan, Y. Li, W. Gao, and J. Li, “On the extremal hyper-wiener index of graphs,” Journal of Chemical and Pharmaceutical Research, vol. 6, no. 3, pp. 477–481, 2014.

W. Gao and L. Shi, “Wiener index of gear fan graph and gear wheel graph,” Asian Journal of Chemistry, vol. 26, no. 11, pp. 3397–3400, 2014.

W. F. Xi and W. Gao, “Geometric-arithmetic index and Zagreb indices of certain special molecular graphs,” Journal of Advances in Chemistry, vol. 10, no. 2, pp. 2254–2261, 2014.

J. Y. Dou, Y. Y. Wang, and W. Gao, “Some characteristics on hyper-wiener index of graphs,” Journal of Chemical and Pharmaceutical Research, vol. 6, no. 5, pp. 1659–1663, 2014.

J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, Berlin , Germany, 2008.

E. Estrada, L. Torres, L. Rodríguez, and I. Gutman, “An atom-bond connectivity index: modelling the enthalpy of formation of alkanes,” Indian Journal of Chemistry A, vol. 37, no. 10, pp. 849–855, 1998.

K. C. Das, I. Gutman, and B. Furtula, “On atom-bond connectivity index,” Chemical Physics Letters, vol. 511, pp. 452–454, 2011.

B. Furtula, A. Graovac, and D. Vukičević, “Atom-bond connectivity index of trees,” Discrete Applied Mathematics, vol. 157, no. 13, pp. 2828–2835, 2009.

T. S. Vassilev and L. J. Huntington, “On the minimum ABC index of chemical trees,” Applied Mathematics, vol. 2, no. 1, pp. 8–16, 2012.

J. Chen, J. Liu, and Q. Li, “The atom-bond connectivity index of catacondensed polyomino graphs,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 598517, 7 pages, 2013.

A. Graovac and M. Ghorbani, “A new version of atom-bond connectivity index,” Acta Chimica Slovenica, vol. 57, no. 3, pp. 609–612, 2010.

M. Rostami, M. S. Haghighat, and M. Ghorbani, “On second atom-bond connectivity index,” Iranian Journal of Mathematical Chemistry, vol. 4, no. 2, pp. 265–270, 2013.

I. Gutman and S. Klavžar, “An algorithm for the calculation of the Szeged index of benzenoid hydrocarbons,” Journal of Chemical Information and Computer Sciences, vol. 35, no. 6, pp. 1011–1014, 1995.

M. V. Diudea, “Phenylenic and naphthylenic tori,” Fullerenes, Nanotubes and Carbon Nanostructures, vol. 10, no. 4, pp. 273–292, 2002.