The Wiener polarity index of benzenoid systems and nanotubes
Niko Tratnik
Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia
niko.tratnik@um.si, niko.tratnik@gmail.com
(Received November 9, 2017)
Abstract
In this paper, we consider a molecular descriptor called the Wiener polarity index, which is defined as the number of unordered pairs of vertices at distance three in a graph. Molecular descriptors play a fundamental role in chemistry, materials engineering, and in drug design since they can be correlated with a large number of physicochemical properties of molecules. As the main result, we develop a method for computing the Wiener polarity index for two basic and most commonly studied families of molecular graphs, benzenoid systems and carbon nanotubes. The obtained method is then used to find a closed formula for the Wiener polarity index of any benzenoid system. Moreover, we also compute this index for zigzag and armchair nanotubes.
Keywords: Wiener polarity index; benzenoid system; zigzag nanotube; armchair nanotube; cut method
AMS Subj. Class: 92E10, 05C12, 05C90
1 Introduction
Benzenoid systems (also called hexagonal systems) represent a mathematical model for molecules called benzenoid hydrocarbons and form one of the most important class of chemical graphs [10]. Similarly, carbon nanotubes are carbon compounds with a cylindrical structure, first observed in 1991 by Iijima [17]. Carbon nanotubes posses many unusual properties, which are valuable for nanotechnology, materials science and technology, electronics, and optics. They can be openended or closedended. Openended singlewalled carbon nanotubes are also called tubulenes. In this paper, we model benzenoid hydrocarbons and tubulenes by graphs.
Theoretical molecular structuredescriptors (also called topological indices) are graph invariants that play an important role in chemistry, pharmaceutical sciences, and in materials science and engineering since they can be used to predict physicochemical properties of organic compounds. The most commonly studied molecular descriptor is the Wiener index introduced in 1947 [31], which is defined as the sum of distances between all the pairs of vertices in a molecular graph. Wiener showed that the Wiener index is closely correlated with the boiling points of alkane molecules. Further work on quantitative structureactivity relationships showed that it is also correlated with other quantities, for example the parameters of its critical point, the density, surface tension, and viscosity of its liquid phase.
The Wiener polarity index of a graph is defined as the number of unordered pairs of vertices at distance three. In the best of our knowledge, Wiener already had some information about the applicability of this molecular descriptor. Later, Lukovits and Linert [24] demonstrated quantitative structureproperty relationships for the Wiener polarity index in a series of acyclic and cyclecontaining hydrocarbons. Also, Hosoya found a physicochemical interpretation for this index, see [13]. Furthermore, the Wiener polarity index is closely related to the Hosoya polynomial [12], since it is exactly the coefficient before in this polynomial. In recent years, a lot of research has been done in investigating the Wiener polarity index of trees and unicyclic graphs, see [1, 7, 8, 16, 22, 23]. Also, the NordhausGaddumtype results for this index were considered in [14, 32]. For some other recent investigations on the Wiener polarity index see [3, 15, 18, 25, 29]. In [2] the Wiener polarity index was expressed by using the Zagreb indices and by applying this result, formulas for fullerenes and catacondensed benzenoid systems were found. In this paper, we choose a different approach and express the index by the number of hexagons and the Wiener polarity indices of smaller graphs.
We proceed as follows. In the following section, some basic definitions and notations are introduced. In Section 3, we develop a cut method for computing the Wiener polarity index of benzenoid systems and tubulenes. For a survey paper on the cut method see [20] and some resent investigations on this topic can be found in [5, 6, 19, 27]. Then, the main result is used to find the closed formulas for the Wiener polarity index of benzenoid systems, zigzag tubulenes, and armchair tubulenes. Finally, some ideas for the future work are presented.
2 Preliminaries
A graph is an ordered pair of a set of vertices (also called nodes or points) together with a set of edges, which are element subsets of . For some basic concepts about graph theory see [30]. Having a molecule, if we represent atoms by vertices and bonds by edges, we obtain a molecular graph. The graphs considered in this paper are always simple and finite. The distance between vertices and of a graph is the length of a shortest path between vertices and in . If there is no confusion, we also write for . Then the Wiener polarity index of a graph , denoted by , is defined as
Let be the hexagonal (graphite) lattice and let be a cycle on it. Then a benzenoid system is induced by the vertices and edges of , lying on and in its interior, see Figure 1. In such a case, every hexagon lying in the interior of is called a hexagon of benzenoid system . The number of all the hexagons of will be denoted by and the number of vertices in cycle is denoted by . Moreover, a vertex of a benzenoid system is called internal if it lies on exactly three hexagons of .
An elementary cut of a benzenoid system is a line segment that starts at the center of a peripheral (boundary) edge of a benzenoid system , goes orthogonal to it and ends at the first next peripheral edge of . In what follows, by an elementary cut we usually mean the set of all edges that intersect the elementary cut. Note that all benzenoid system are also partial cubes, which are defined as isometric subgraphs of hypercubes (a subgraph of is isometric if for any two vertices it holds ) and represent a large class of graphs with a lot of applications. In particular, every elementary cut of a benzenoid system coincides with a class. Recall that two edges and of a connected graph are in relation , , if and that in a partial cube this relation is always transitive. For more details about relation and partial cubes see [11].
Next, we define openended carbon nanotubes, also called tubulenes (see [26]), which represent an important family of chemical graphs. Choose any lattice point in the infinite and regular hexagonal lattice as the origin . Moreover, let be a point in the hexagonal lattice such that the graph distance between and is an even number greater or equal to six. In addition, let and be the two basic lattice vectors (see Figure 2). Obviously, there are integers such that . Draw two straight lines and passing through and perpendicular to , respectively. By rolling up the hexagonal strip between and and gluing and such that and superimpose, we can obtain a hexagonal tessellation of the cylinder. and indicate the direction of the axis of the cylinder. Using the terminology of graph theory, a tubulene is defined to be the finite graph induced by all the hexagons of that lie between and , where and are two vertexdisjoint cycles of encircling the axis of the cylinder. Any such hexagon (between and ) is called a hexagon of a tubulene and the number of these hexagons will be denoted by . The vector is called a chiral vector of and the cycles and are the two openends of . For any tubulene , if its chiral vector is , is called an type tubulene, see Figure 2.
Remark 2.1
Sometimes, the definition of a tubulene does not contain the requirement that the graph distance between and is at least six (but very often, some other condition on and is added). However, if this distance is four (or even less), it can happen that two distinct hexagons have two common edges, which is not usual for the considered molecules. Moreover, by including such a requirement, a tubulene can not contain a cycle of length four (or less), which is essentially used in Theorem 3.5.
3 A cut method for benzenoid systems and tubulenes
To develop a cut method for the Wiener polarity index, some definitions and lemmas are first needed. Obviously, in the regular hexagonal lattice there are exactly three directions of edges. Let , , and be the sets of edges of the same direction. Moreover, if is a benzenoid system or a tubulene (drawn in the hexagonal lattice), we define and often use instead of , where . One can notice that for a benzenoid system the set is the set of all elementary cuts in a given direction. For our consideration it is important to observe that for any benzenoid system or a tubulene it holds:

any two hexagons of have at most one edge in common (see Remark 2.1),

on any hexagon of there are exactly two edges from , two edges from , and two edges from ,

if , , are two distinct edges, then they have no vertex in common.
In the rest of the paper we will denote by , , the graph obtained from by deleting all the edges from . Also, for let (or simply ) be the set of all connected components of the graph . Furthermore, let .
Lemma 3.1
If is a benzenoid system, then every connected component of , , is a path on at least three vertices.
Proof. It is obvious that every connected component is a path. Since every elementary cut goes through two opposite edges of a hexagon, every component must have at least three vertices.
However, in the case of tubulenes a connected component can be a cycle (see, for example, zigzag tubulenes in Section 5).
Lemma 3.2
If is a tubulene, then every connected component of , , is a path or a cycle.
Proof. All vertices of degree three in have exactly one incident edge in for every . Therefore, after deleting all the edges of from , every vertex of has degree at most two. Therefore, every connected component of is a path or a cycle.
A hexagon of the hexagonal lattice is called external for a benzenoid system if is not a hexagon of but . Similarly, a hexagon of the hexagonal tessellation is called external for a tubulene if is not a hexagon of but . If is an external hexagon of a benzenoid system or a tubulene , we notice that a largest connected component of the intersection is always a path. Especially important are external hexagons for which such a component is a path on at least four vertices. Therefore, the set of all external hexagons for which the largest connected component of is a path will be denoted as . Also, the set (or ) is defined as the set of all external hexagons of for which the largest connected component of is a path (or ). Moreover, we denote the cardinality of by , i.e. for .
In Figure 3 we can see a benzenoid system , where hexagons of are coloured grey and external hexagons of are white. Obviously, , , and .
In order to develop a cut method for the Wiener polarity index, we next study the structure of shortest paths of length three.
Lemma 3.3
Let be a benzenoid system or a tubulene and let be a shortest path of length three in such that for any . Then all vertices and edges of lie on a hexagon of or on an external hexagon of .
Proof. Denote the edges of by , , and such that is incident with and is incident with . Without lost of generality assume that for any . Obviously, there is exactly one hexagon , which is a hexagon of or an external hexagon of , such that and belong to . Let be an endpoint of such that is not an endpoint of and let be an edge of such that is incident with and . Since edges of any hexagon alternatively belong to , , and , it follows that . Therefore, since no two edges from are incident, it follows that and the proof is complete.
Lemma 3.4
Let be a benzenoid system or a tubulene and let be a shortest path of length three in . Then exactly one of the following two statements holds:

belongs to exactly one hexagon, which is a hexagon of or an external hexagon of ,

there exists exactly one such that belongs to exactly one connected component of .
Proof. If two edges and are incident, they can not both belong to the set , . Therefore, the edges of can not all belong to one set . Hence, we have two possibilities:

If for any , then by Lemma 3.3 belongs to (exactly) one hexagon of or to one external hexagon of .

If there exists such that (note that such is unique), then all vertices and edges of belong to the graph . Since is connected, it belongs to exactly one component of .
Therefore, the proof is complete.
Finally, we are able to prove the main result of this section, which enables us to compute the Wiener polarity index of a benzenoid system or a tubulene.
Theorem 3.5
Let be a benzenoid system or a tubulene. Then
Proof. Denote by the set of all unordered pairs of vertices such that lie on the same hexagon of and . Similarly, denote by the set of all unordered pairs of vertices such that lie on the same external hexagon of and . Moreover, let be the set of all unordered pairs of vertices such that and do not lie on the same hexagon. It follows
Obviously, for any hexagon of , there are three unordered pairs of vertices on at distance three. On the other hand, if for two vertices on the same hexagon it holds , then by Lemma 3.4 this hexagon is unique. Therefore, .
If , there is exactly one pair of vertices at distance three. If we have two such pairs and for there are exactly three such pairs. Hence,
If are vertices at distance three that do not lie on the same hexagon, then by Lemma 3.4 they belong to exactly one connected component of , where . For the contrary, let belong to exactly one connected component of for some and . We will show that . Consider two cases:

If is a benzenoid system, it obviously follows since there is only one shortest path between and in .
Since and belong to one connected component of , they can not both belong to one hexagon. Hence, and we are finished.
The previous result enables us to reduce the problem of computing the Wiener polarity index of to the problem of computing the Wiener polarity indices of paths or cycles, which is trivial. In particular, if is a path on vertices, then
(1) 
and if is a cycles on vertices, we have
(2) 
4 A closed formula for benzenoid systems
In this section we focus on benzenoid system and use the developed cut method to obtain a closed formula for the Wiener polarity index of an arbitrary benzenoid system. Throughout the section we denote by , , the number of elementary cuts of a benzenoid system that contain only edges from (the elementary cuts in a given direction). First, three auxiliary results are needed.
Lemma 4.1
[9] Let be a benzenoid system with internal vertices. Then .
Lemma 4.2
Let be a benzenoid system with the boundary cycle . Then
Proof. Obviously, every elementary cut of intersects the boundary cycle exactly twice. Since the number of elementary cuts is , we have that the number of edges in is . Hence, the result follows.
Lemma 4.3
Let be a benzenoid system. Then the graph , , has exactly connected components.
Proof. When all the edges from one elementary cut are deleted, the obtained graph has exactly two connected components. Therefore, if we delete elementary cuts one by one, at every step we get one additional component. After deleting all the elementary cuts we have connected components.
Note that Lemma 4.3 also follows from the fact that the quotient graph , which has connected components of as vertices, two such components being adjacent whenever there is an edge in connecting them, is always a tree [4]. In such a tree, every vertex represents a connected component of and every edge represents an elementary cut in direction . For the details see [4].
Finally, we are able to show the main result of this section.
Theorem 4.4
Let be a benzenoid system. Then
Proof. Let . Since by Lemma 4.3 the graph has exactly connected components, we denote by the number of vertices in the connected components of . Obviously
(3) 
By Theorem 3.5,
(4) 
By inserting this into Equation 4 we deduce
and then, using Lemma 4.2 one can obtain
Obviously, the number of internal vertices of is exactly . Hence, Lemma 4.1 implies . Inserting this in the previous equation we finally get
which finishes the proof.
Note the the number of hexagons of a benzenoid system can be obtained from its boundary edges code (see [21] for the details). Therefore, by Theorem 4.4 the Wiener polarity index of a benzenoid system can be determined by the shape of its boundary. To show how Theorem 4.4 can be used, we demonstrate it on one simple example. Let be a benzenoid system from Figure 3. We can immediately see that and . Therefore, by Theorem 4.4 we obtain
5 The Wiener polarity index of zigzag and armchair tubulenes
The aim of this section is to use the obtained cut method to calculate closed formulas for the Wiener polarity index of zigzag and armchair tubulenes.
5.1 Zigzag tubulenes
If is a type tubulene where or , we call it a zigzag tubulene. Let be a zigzag tubulene such that are the shortest possible cycles encircling the axis of the cylinder (see Figure 4). If has layers of hexagons, each containing exactly hexagons, then we denote it by , i.e. . We always assume and (see Remark 2.1).
Obviously, the graph has connected components, each isomorphic to a cycle on vertices. Moreover, the graph has connected components, each isomorphic to a path on vertices, and the same holds for the graph . Also, we notice that . Therefore, by Theorem 3.5 we obtain
5.2 Armchair tubulenes
If is a type tubulene where , we call it an armchair tubulene. Let be an armchair tubulene such that and are the shortest possible cycles encircling the axis of the cylinder and such that there is the same number of hexagons in every column of hexagons (see Figure 5). If has vertical columns of hexagons, each containing exactly hexagons, then we denote it by , i.e. . Obviously, must be an even number. Note that is a type tubulene. Furthermore, we always assume and (see Remark 2.1).
First, we observe and . Also, it is not difficult to see that the graph has exactly connected components, all isomorphic to a path on at least three vertices, see Figure 6.
Denote the number of vertices in these components by . Therefore, by Equation 1
and the same result holds for . On the other hand, the graph has exactly connected components, each isomorphic to a path on vertices. Therefore, by Equation 1
Finally, using Theorem 3.5 we conclude
6 Concluding remarks
In the paper we have developed a method for computing the Wiener polarity index of benzenoid system and tubulenes. Naturally, one can ask to which other planar or molecular graphs a similar method can be applied. Furthermore, in the case of benzenoid systems the elementary cuts play an important role. Therefore, since all benzenoid systems are partial cubes (and in partial cubes elementary cuts represent classes), it would be interesting to find a generalization of the described method to all partial cubes.
Acknowledgment
The author was financially supported by the Slovenian Research Agency.
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