On the Extremal Wiener Polarity Index of Hückel Graphs

Graphs are used to model chemical compounds and drugs. In the graphs, each vertex represents an atom of molecule and edges between the corresponding vertices are used to represent covalent bounds between atoms. The Wiener polarity index W p(G) of a graph G is the number of unordered pairs of vertices u, v of G such that the distance between u and v is equal to 3. The trees and unicyclic graphs with perfect matching, of which all vertices have degrees not greater than three, are referred to as the Hückel trees and unicyclic Hückel graphs, respectively. In this paper, we first consider the smallest and the largest Wiener polarity index among all Hückel trees on 2n vertices and characterize the corresponding extremal graphs. Then we obtain an upper and lower bound for the Wiener polarity index of unicyclic Hückel graphs on 2n vertices.


Introduction
Nearly half a century ago, the development of quantum chemistry is largely due to the wide application of the concept of graph. One of the major topics in this field is molecular topological index. The molecular topological index can describe the structure of the molecule quantitatively, as an invariant of the graph can be used to demonstrate the relationship between the molecules structure and performance. Quantitative structure activity relationships are a popular computational biology paradigm in modern drug design.
One of the most widely known topological descriptors is Wiener polarity index. The Wiener polarity index of an organic molecule graph of which = ( ( ), ( )) is defined by which is the number of unordered pairs of vertices , V of such that ( , V) = 3, where ( , V) denotes the distance between two vertices and V in . The Wiener polarity index for the quantity defined in the equation above is introduced by Wiener [1] for acyclic molecules in a slightly different yet equivalent manner. Moreover, Wiener [1] used a linear formula for the Wiener index fl ∑ ,V⊆ ( , V) and the Wiener polarity index to calculate the boiling points of the paraffins; that is, where , , and are constants for a given isomeric graph. In 1998, by using the Wiener polarity index, Lukovits and Linert [2] demonstrated quantitative structure-property relationships in a series of acyclic and cycle-containing hydrocarbons. Besides, a physical-chemical interpretation of ( ) was found by Hosoya [3]. Recently, Du et al. [4] obtained the smallest and largest Wiener polarity indices together with the corresponding graphs among all trees on vertices, respectively. Deng et al. [5] characterized the extremal Wiener polarity index of trees with a given diameter. The authors in [6] found the maximum Wiener polarity index among all chemical trees with vertices and pendents. Hou et al. [7] found the maximum Wiener polarity index of unicyclic graphs together with the corresponding extremal graphs.
As is well known, conjugated hydrocarbon molecules considered in the Hückel molecule orbit theory are usually represented by the carbon-atom skeleton graphs, of which all vertices have degrees less than four. We call such molecular graphs Hückel molecular graphs. In graph theory, the Hückel molecular graphs with Kekulé structures are graphs with perfect matchings of which the largest degree of vertices does not exceed three. ( ) be the neighbor vertex set of in . Then ( ) = | ( )| is called the degree of . If (V) = 1, then we call V a pendent vertex of . Let Δ( ) denote the maximum vertices degree in . As usual, let and be the cycle and path of order , respectively. A path in is called -degree pendent chain if all its internal vertices are of degree 2 and its ends of degrees 1 and , respectively, where ≥ 3. A matching of the graph is a subset of ( ) such that no two edges in share a common vertex. If is a matching of a graph and vertex V is incident with an edge of , then V is said to be -saturated, and if every vertex of is -saturated, then is a perfect matching. Suppose V ∈ ( ); the notion − V denotes the new graph yielded from by deleting the edge V. Similarly, if V ∉ ( ), then + V denotes the new graph obtained from by adding the edge V. The set of the Hückel trees and Hückel unicyclic graphs with 2 vertices is denoted by T 2 and H 2 , respectively.
In the paper, we consider the Wiener polarity index for Hückel trees and Hückel unicyclic graphs. In Section 2, we discuss some properties of the Wiener polarity index of Hückel trees. In Section 3, we determine the smallest and largest Wiener polarity index together with the corresponding graphs among all Hückel trees. In Section 4, the smallest and the largest Wiener polarity indices among all Hückel unicyclic graphs on 2 vertices are identified, respectively.

Some Properties of the Wiener Polarity Index of Hückel Trees
In this section, first, we give some formulas for computing the Wiener polarity index of trees.
Lemma 1 (see [4]). Let = ( , ) be a tree. Then Lemma 2 (see [8]). Let be a 2 -vertex tree ( ≥ 2) with a perfect matching. Then has at least two pendent vertices such that each is adjacent to vertices of degree two.
For any ∈ T 2 , the following several lemmas will give necessary conditions on which ( ) attains the maximum values.

Lemma 3.
Let be a graph in T 2 such that ( ) is as larger as possible. Then the lengths of all pendent chains in are no more than 2.
Proof. By contradiction. Assume that there exists a pendent chain = 0 1 2 ⋅ ⋅ ⋅ −1 with length such that ≥ 3; we distinguish the following two cases.
Case 1 ( = 3). This implies that there exists a pendent chain = 0 1 2 3 such that ( 0 ) = 3, ( 1 ) = ( 2 ) = 2, and ( 3 ) = 1. We claim that the vertex adjacent to 0 cannot be a pendent vertex; suppose, on the contrary, that V 0 is a pendent vertex adjacent to 0 . Assuming that is the perfect matching of , we know that is unique in trees and each pendent edge of belongs to ; therefore, 0 V 0 ∈ and 2 3 ∈ . Then 1 is not saturated by , a contradiction. Let V be a vertex of degree 2 nearest to 0 except for 1 . Let = − 1 2 + V 2 , then obviously, ∈ T 2 . The following two subcases should be considered.
In this case, we assume that is another neighbor of V; by Lemma 1, we have It contradicts the maximality of ( ).
By Lemma 3, we can show that if ∈ T 2 with maximum ( ), the length of any pendent chain is either 2 or 1. Therefore, we have reduced the problem to the Hückel trees having a path with both ends of degree 3. Then, we introduce a graph transformation which will be used in the following proof.
Let be a tree in T 2 with ≥ 2. Let = V be a nonpendent edge of . 1 and 2 are two components of − , ∈ 1 , and V ∈ 2 . 0 is the graph obtained from in the following way: (1) Contract the edge = V (i.e., identify of 1 with V of 2 ).
(2) Add a pendent edge to the vertex (= V).
We call procedures (1) and (2) the edge-growth transformation of or e.g.t of for short (see Figure 1).

Lemma 4.
Let be a graph in T 2 such that ( ) is as larger as possible. If is a path in with two end-vertices of degree 3, then all internal vertices of are of degree 3. Proof. Suppose, on the contrary, that there is a path V 1 ⋅ ⋅ ⋅ V in such that ( ) = 3, ( ) = 3, (V 1 ) = ⋅ ⋅ ⋅ = (V ) = 2, and ≥ 1. Let be the perfect matching of ; we consider the following two cases.
Case 1 ( is even). In this case, it is easy to see that either If not, there must exist a vertex of path V 1 ⋅ ⋅ ⋅ V not saturated by . We distinguish the following subcases.
One can transform into by using exactly /2 steps of e.g.t for above edges continuously; we note that the resulting graph is a tree obtained by attaching one pendent edge to each vertex of V 1 , V 3 , . . . , V −1 . Then ∈ T 2 . Then by Lemma 1, we have which contradicts the maximality of ( ).
. In this subcase, we can easily see that the vertices V 2 , V 3 , V 4 , . . . , V −2 , V −1 of are mutually matched. That is to say, ; then one can transform into by using exactly ( − 2)/2 steps of e.g.t continuously. We notice that the resulting graph is a Hückel tree obtained by attaching one pendent edge to each vertex of V 2 , V 4 , . . . , V −2 . Then is one of class (I) of trees, as shown in Figure 2.
Let and be two neighbors of vertex and and be two neighbors of , respectively. Let 1 , 2 , 3 , and 4 be the connected components containing , , , and of the graph − − , respectively. Also, by Lemma 1, we have If > 3, ( ) > ( ). If = 2, is also one of class (I) of trees.
In the following, we have reduced the problem to the Hückel trees of class (I). For V 1 ∈ , then and cannot be pendent edges of , since ∈ T 2 . There are at least two vertices in 1 and 2 ; without loss of generality, we consider 1 . Let ( ) be the number of vertices of . We distinguish the following subcases.  to be the perfect matching of ; then = − V 1 − + + V 1 is the perfect matching of ; we notice that ∈ T 2 , and by Lemma 1, there is From Lemma 3, it is noted that ( ) = 3; otherwise, if ( ) = 2, then there exists a pendent chain with length of at least 3, a contradiction. Let 1 and 2 denote the two neighbors of in 1 ; it should be noted that 1 ≤ ( 1 ) ≤ 3 and 1 ≤ ( 2 ) ≤ 3. Then by Lemma 1, we have Therefore, ( ) > ( ) > ( ), a contradiction. The analysis on V of degree 2 is the same as that for V 1 .
Case 2 ( is odd). In this case, there are odd vertices of degree 2 in the path V 1 ⋅ ⋅ ⋅ V ; then there exits exactly one of two edges V 1 and V which belongs to ; without loss of generality, we assume that V 1 ∈ . It should be noted that V 2 V 3 ∈ and V 4 V 5 ∈ , . . . , V −1 V ∈ ; then one can transform into by using exactly ( −1)/2 steps of e.g.t continuously. We notice that the resulting graph is a Hückel tree obtained by attaching one pendent edge to each vertex of V 2 , V 4 , . . . , V −1 ; then is one of class (II), as shown in Figure 3. We also notice that If > 2, there is ( ) > ( ). If = 1, is also in class (II) of trees. The proof is similar to that of Subcase 1.2.
In any case, the resulting graph belongs to T 2 such that all the internal vertices of the path V 1 ⋅ ⋅ ⋅ V are of degree 3. Furthermore, the resulting graph has the value of Wiener polarity no less than that of , which contradicts the maximality of ( ). This completes the proof.
The next result follows obviously from the proof of Lemmas 3 and 4. (iii) All the vertices of degree 2 in * are on the pendent chains.

The Extremal Wiener Polarity Index of Hückel Trees
In this section, we will discuss the maximum and minimum Wiener polarity index of Hückel trees with 2 vertices. Firstly, we consider the Hückel trees with the largest Wiener polarity index. Let be the number of edges in between vertices of degrees and . By Lemma 1, we have In particular, if is a Hückel tree, then Let * ∈ T 2 with a vertices sequence ( 1 , 2 , 3 ), where denotes the number of vertices of * with degree . Recall the following relations: (15) From above two equalities it follows that By Corollary 5, it should be noted that the subgraph induced by the vertices of degree 3 in * is also a tree. Then we deduce that Then, by Corollary 5, we have By above equations, we have that ( * ) = 2(2 − 2 3 − 2) + 4( 3 −1) = 4 −8. From Corollary 5 and the arguments above, the following result is obvious. Next, we consider the minimum Wiener polarity index among T 2 , and we first consider some special cases. If = 1, 2 ≅ 2 and ( 2 ) = 0; If = 2, 2 ≅ 4 and ( 4 ) = 1. In the following, we assume that ≥ 3. For all Hückel trees in T 2 , sharp lower bounds for ( ) are obtained in the following theorem. Proof. We prove the assertion by induction on . If = 3, then ≅ 6 or ≅ 1 (see Figure 4). It can be easily checked that ( 6 ) = 3 < ( 1 ) = 4. The result holds for = 3. Now assume the assertion holds for all Hückel trees with less than ≥ 4 vertices. Suppose is a Hückel tree with 2 vertices; by Lemma 2, then there exists a pendent vertex V which is adjacent to of degree 2. Let = − −V; it should be noted that ∈ T 2 −2 . Let be adjacent to of ; that is, ∈ ( ). We distinguish the following cases.

The Wiener Polarity Index of Unicyclic Hückel Graphs
In this section, we will give sharp lower and upper bounds for Wiener polarity index of unicyclic Hückel graphs. The girth ( ) of a connected graph is the length of shortest cycle in .
First, we will establish some lemmas which will be useful to the proofs of our main results.
Proof. We only prove the first assertion, and the second assertion can be proved analogously. Let be a unicyclic Hückel graph with ( ) = 3 and V, , and be the three vertices on the unique cycle of , and let (V) = 1 + 2, ( ) = 2 + 2, and ( ) = 3 + 2.
Let be the perfect matching of ; there exists one edge on the unique cycle of that does not belong to ; otherwise, there is a contradiction to the fact that has perfect matching; without loss of generality, suppose that ∉ , and then deleting the edge , we get a Hückel tree and still is the perfect matching of . By Lemmas 1 and 8, we have Since is a unicyclic Hückel graph, without loss of generality, we may assume that 0 ≤ 2 ≤ 3 ≤ 1. Then Hence, by Theorem 6, we obtain that Similarly, by Theorem 7, we obtain that (3) if ( ) = 5, then 2 − 7 ≤ ( ) ≤ 4 − 1.
Proof. We only prove the first assertion and other assertions can be proved similarly. Let be a unicyclic Hückel graph with ( ) ≥ 7. Let be the perfect matching of ; then there exists edge V ∈ ( ) such that V ∉ . We can get a Hückel tree by deleting V. By Lemma 8, we have This completes the proof.
Combining Lemmas 9 and 10, we have the following result.

Conclusion
This paper determined the smallest and the largest Wiener polarity index among all Hückel trees and unicyclic Hückel graphs on 2 vertices and characterized the corresponding extremal graphs. Thus, the promising prospects of the application for the chemical and pharmacy engineering will be illustrated in the theoretical conclusion that is obtained in this paper.