The inverse wiener polarity index problem for chemical trees national gas average 2012

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 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]:

Here, it should be mentioned that W p is the same as the reduced second Zagreb index [ 16– 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– 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– 34] and related references listed therein. Li et al. [ 35] addressed the inverse problem based on four other well-known 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 ∈ { n − 3, n − 2,…,3 n − 16, 3 n − 15}, where n ≥ 6, there exists an n-vertex chemical tree T such that W p( T) = t.

For Case 2, since the order is 3 k + 1, adding one more pendent vertex to x is enough for us to form chemical trees of order 3 k + 1 with desired Wiener polarity indices. While in Case 3, note that the order is 3 k + 2, we need to attach a pendent vertex to x and a pendent vertex to y to obtain our desired chemical trees.

Until now, all the chemical trees with desired Wiener polarity indices are constructed, except when t = n(= 3 k + 1). Aimed to this remaining case, we review the chemical tree of order 3 k constructed in Case 1 with Wiener polarity index 3 k (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 = 3 k + 1 with Wiener polarity index n = 3 k + 1.

But this time, we need add two more vertices. After attaching a pendent vertex to x and a pendent vertex to y, and using Lemma 3 twice, this operation increase the Wiener polarity index by 6, which implies that it results in a series of chemical trees of order n = 3 k + 2 with Wiener polarity indices

For the remaining cases t = n(= 3 k + 2), n + 1(= 3 k + 3) and n + 2(= 3 k + 4), recall that each of the chemical trees of order 3 k constructed in Case 1 with Wiener polarity indices 3 k, 3 k + 1, 3 k + 2 (i.e., the last chemical trees in Figs 4, 5 and 6) contains a vertex of degree 2 with pendent neighbor, say z. Here by applying Lemma 3 twice, attaching a path on two vertices to z would increase the Wiener polarity index by 2. Therefore we may construct three chemical trees of order n = 3 k + 2 with Wiener polarity indices n = 3 k + 2, n + 1 = 3 k + 3 and n + 2 = 3 k + 4, respectively.

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