On the maximal Wiener index and related questions

The Wiener index of a graph is the sum of the distances between all pairs of vertices. It has been one of the main descriptors that correlate a chemical compound's molecular structure with experimentally gathered data regarding the compound's characteristics. In 2008, Wang and Zhang independently characterized trees with specified degree sequence that minimize the Wiener index. In the paper of Wang, a corollary on maximizing the Wiener index was pointed out to be incorrect by Zhang et. al. in 2010. Zhang et. al. also provided partial results and noted that the question turns out to be complicated. Later, ela et. al. considered this question as a quadratic assignment problem and provided a polynomial time algorithm. We make some progress in this contribution, providing information on the candidate trees for the maximum Wiener index. Some interesting combinatorial relations to other objects arose from this study. We also consider the bound of this maximum value as well as study this question for trees with small diameter and for chemical trees with specified degree sequence.