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 W-p) was devised by the chemist Harold Wiener, for predicting the boiling points of alkanes. The index W-p 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 is an element of {n - 3, n - 2, . . .,3n - 16, 3n - 15}, n >= 6, there exists an n-vertex chemical tree T such that W-p(T) = t.