The inverse Wiener polarity index problem for chemical trees

The Wiener polarity number (which, nowadays, known as the Wiener polarity index and usually denoted by Wp) was devised by the chemist Harold Wiener, for predicting the boiling points of alkanes. The index Wp of chemical trees (chemical graphs representing alkanes) is defined as the number of unordered pairs of vertices (carbon atoms) at distance 3. The inverse problems based on some well-known topological indices have already been addressed in the literature. The solution of such inverse problems may be helpful in speeding up the discovery of lead compounds having the desired properties. This paper is devoted to solving a stronger version of the inverse problem based on Wiener polarity index for chemical trees. More precisely, it is proved that for every integer t ∈ {n − 3, n − 2,…,3n − 16, 3n − 15}, n ≥ 6, there exists an n-vertex chemical tree T such that Wp(T) = t.


Introduction
A (chemical) topological index is a real number calculated from chemical graphs (graphs representing chemical compounds, in which vertices represent atoms and edges represent covalent bonds between atoms) such that it remains unchanged under graph isomorphism [1]. Topological indices are usually used in quantitative structure-activity and structure-property relationships studies for predicting the biological activities or physical-chemical properties of chemical compounds [2].
The Wiener polarity number (which, nowadays, known as the Wiener polarity index and usually denoted by W p ) was devised in 1947 by the chemist Harold Wiener [3] for predicting the boiling points of alkanes, and this index is among the oldest topological indices. The index W p of chemical trees (chemical graphs representing alkanes) is defined as the number of unordered pairs of vertices at distance 3.
Lukovits and Linert [4] extended the definition of W p for cycle-containing chemical graphs by using a heuristic approach, and used this new definition to demonstrate quantitative structure-property relationships in a series of acyclic and cycle-containing hydrocarbons. Hosoya and Gao [5] found that the relative magnitude of W p among isomeric alkanes keeps pace with the number of gauche structures in the most probable confirmation, and thus W p can predict the relative magnitude of liquid density. Miličević and Nikolić [6] used W p in modeling the a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 boiling points of lower (C 3 -C 8 ) alkanes. Shafiei and Saeidifar [7] performed quantitative structure-activity relationships studies on 41 sulfonamides for predicting their heat capacity and entropy, using W p together with some other topological indices. In a recent study [8], some models for predicting the thermal energy, heat capacity and entropy of 19 amino acids were developed and it was found that W p is a good topological index for modeling thermal energy.
In this paper, we are concerned with the possible values of W p for chemical trees. As usual, denote by uv the edge connecting the vertices u, v in a chemical tree T, and d T (u) the degree of vertex u in T. The following beautiful result is due to Du et al. [15]: Lemma 1. Let T be a (chemical) tree. Then where E(T) denotes the edge set of T.
Here, it should be mentioned that W p is the same as the reduced second Zagreb index [16][17][18] in case of (chemical) trees. Deng [19] reported the maximum W p value of chemical trees. The same authors of this paper [20] characterized all the chemical trees with maximum W p value. Recently, Ashrafi and Ghalavand [21] determined the first two minimum W p values of chemical trees and characterized the corresponding chemical trees attaining the first two minimum W p values. In the reference [22], main extremal results of the paper [21] are re-established in an alternative but shorter way, and all members with the third minimum W p value are determined from the collection of all n-vertex chemical trees.
The problem of finding chemical structure(s) corresponding to a given value of a topological index TI is known as the inverse problem based on TI [23]. Solutions of such inverse problems may be helpful in designing a new combinatorial library, and speed up the discovery of lead compounds with some desired properties [24].
Study of the inverse problem based on topological indices was initiated by the Zefirov group in Moscow [25][26][27][28][29]. Gutman [30] studied the inverse problem based on the Wiener index (this index appeared in the same paper [3] where W p was reported, see the recent survey [31] for more details about Wiener index). Solving the inverse problem based on Wiener index was the subject of several papers, for example see the papers [32][33][34] and related references listed therein. Li et al. [35] addressed the inverse problem based on four other wellknown topological indices, introduced in mathematical chemistry. Recently, an inverse problem based on the k-th Steiner Wiener index (a generalized version of Wiener index) was studied in the paper [36]. Further details about inverse problem can be found in the survey [37], recent papers [38,39] and related references listed therein.
Here we attempt to solve a stronger version of the inverse problem based on Wiener polarity index for chemical trees. We have been able to show that for every integer t 2 {n − 3, n − 2,. . .,3n − 16, 3n − 15}, where n ! 6, there exists an n-vertex chemical tree T such that W p (T) = t.

Methods
By Lemma 1, we may get the following two lemmas immediately.
Lemma 2. Let T and T 1 be the two chemical trees as depicted in Fig 1. Then In particular, the transformation from T to T 1 depicted in Lemma 2 is called a grafting pendent path transformation at v in T.
A vertex of degree 1 is said to be a pendent vertex. Lemma 3. Suppose that v is a pendent vertex with unique neighbor u in the chemical tree T. Let T 1 be another chemical tree obtained from T by attaching a pendent vertex to v. Then

Results
Theorem 1. For every integer n − 3 t 3n − 15, where n ! 6, there exists a chemical tree T of order n such that W p (T) = t, i.e., fW p ðTÞ : T is a chemical tree of order ng ¼ fn À 3; n À 2; . . . ; 3n À 16; 3n À 15g: Proof. Clearly, the three chemical trees of order n depicted in Fig 2 have Wiener polarity indices n − 3, n − 2 and n − 1, respectively. So we need only to focus on the existence of chemical trees T of order n with W p (T) = t, where n t 3n − 15, i.e., n ! 8.
For the case n = 8, t can only be 8 or 9. It is easily checked that the chemical tree of order 8 obtained from P = v 1 v 2 v 3 v 4 v 5 by attaching three pendent vertices each to v 2 , v 3 , v 4 has Wiener polarity index 8, and the chemical tree of order 8 obtained from P = v 1 v 2 v 3 v 4 v 5 by attaching a pendent vertex to v 2 and two pendent vertices to v 3 has Wiener polarity index 9.
Suppose in the following that n ! 9. We partition our proof into three cases according to the value n(mod 3). Case 1. n = 3k, where k ! 3.
For the three chemical trees T 1 , T 2 and T 3 of order n = 3k in Fig 3, it is easily verified that First, we start with the chemical tree T 1 as depicted in Fig 3, whose Wiener polarity index is 9k − 15. We apply grafting pendent path transformations successively at which gives 2k − 5 transformations in total. A detailed illustration can be found in Fig 4. In particular, for the above series of grafting pendent path transformations, by Lemma 2, the Wiener polarity index would decrease 3 each time. This means that we may construct a respectively.
Next, the initial tree is changed as the chemical tree T 2 in Fig 3, its Wiener polarity index is 9k − 16. This time, we will use grafting pendent path transformations successively at  respectively.
Combining the above arguments, we get a series of chemical trees of order n = 3k with Wiener polarity indices 3k À 3; 3k À 2; . . . ; 9k À 16; 9k À 15; or equivalently, n À 3; n À 2; . . . ; 3n À 16; 3n À 15: Before continuing our proofs for Cases 2 and 3, we first sketch our strategy. From Case 1, a series of chemical trees of order 3k with Wiener polarity indices 3k À 3; 3k À 2; . . . ; 9k À 16; 9k À 15 have been constructed. Notice that each of the chemical trees of order 3k constructed in Case 1 (see Figs 4, 5 and 6) with Wiener polarity indices 3k À 1; 3k; . . . ; 9k À 16; 9k À 15 has some vertex of degree 4 with two pendent neighbors, say x, y. For Case 2, since the order is 3k + 1, adding one more pendent vertex to x is enough for us to form chemical trees of order 3k + 1 with desired Wiener polarity indices. While in Case 3, note that the order is 3k + 2, we need to attach a pendent vertex to x and a pendent vertex to y to obtain our desired chemical trees. Case 2. n = 3k + 1, where k ! 3. In this case, as previous arguments, starting from the chemical trees of order 3k with Wiener polarity indices 3k À 1; 3k; . . . ; 9k À 16; 9k À 15; after adding one more pendent vertex to x, from Lemma 3, such operation increases the Wiener polarity index by 3, so we would get a series of chemical trees of order n = 3k + 1 with Wiener polarity indices 3k þ 2; 3k þ 3; . . . ; 9k À 13; 9k À 12; or equivalently, n þ 1; n þ 2; . . . ; 3n À 16; 3n À 15: The inverse Wiener polarity index problem for chemical trees Until now, all the chemical trees with desired Wiener polarity indices are constructed, except when t = n(= 3k + 1). Aimed to this remaining case, we review the chemical tree of order 3k constructed in Case 1 with Wiener polarity index 3k (i.e., the last chemical tree in Fig  4), obviously it consists of a vertex of degree 2 with pendent neighbor, say u. By Lemma 3, attaching a pendent vertex to u would increase its Wiener polarity index by 1, i.e., we may construct a chemical tree of order n = 3k + 1 with Wiener polarity index n = 3k + 1.
The proof is completed.
To illustrate our main result, let us consider an example for n = 9. Example 1. If n = 9, then from Fig 7, it is clear that for every integer n − 3 = 6 t 12 = 3n − 15, there exists a chemical tree T of order 9 such that W p (T) = t, and hence fW p ðTÞ : T is a chemical tree of order 9g ¼ f6; 7; . . . ; 12g: The inverse Wiener polarity index problem for chemical trees

Discussion
In this paper, we prove that the Wiener polarity indices of chemical trees are continuous, that is to say, there is no gap between the minimum value n − 3 and the maximum value 3n − 15 for the Wiener polarity indices of n-vertex chemical trees. As a consequence, we may get a full ordering for the Wiener polarity indices of chemical trees, which extends the ordering about the first three minimum Wiener polarity indices of chemical trees obtained in [21,22], and the maximum Wiener polarity index of chemical trees obtained in [19,20].