A Conjecture on Wiener Indices in Combinatorial Chemistry

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Drugs and other chemical compounds are often modeled as polygonal shapes, where each vertex represents an atom of the molecule, and covalent bonds between atoms are represented by edges between the corresponding vertices. This polygonal shape derived from a chemical compound is often called its molecular graph, and can be a path, a tree, or in general a graph. An indicator defined over this molecular graph, the Wiener index, has been shown to be strongly correlated to various chemical properties of the compound. The Wiener index conjecture for trees states that for any integer n (except for a finite set), one can find a tree with Wiener index n. This conjecture has been open for quite some time, and many authors have presented incremental progress on this problem. In this paper we present further progress towards proving this conjecture—through the design of efficient algorithms, we show that enumerating all possible trees to verify this conjecture (as done by all the previous approaches) is not necessary, but instead searching in a small special family of trees suffices, thus achieving the first polynomial (in n) time algorithm to verify the conjecture up to integer n. More precisely, we (i) present an infinite family of trees and prove various properties of these trees, (ii) show that a large number of integers, up to at least 108 (compared with the previous best 104) are representable as Wiener indices of trees in this succinct family, (iii) provide several efficient algorithms for computing trees with given Wiener indices, and (iv) implement our algorithms and experimentally show that their performance is asymptotically much better than their theoretical worst-case upper bound.