On the Parity of the Wiener Index of Trees
European Journal of Combinatorics
It is a known fact that the Wiener index (i.e. the sum of all distances between pairs of vertices in a graph) of a tree with an odd number of vertices is always even. In this paper, we consider the distribution of the Wiener index and the related tree parameter “internal path length” modulo 2 by means of a generating functions approach as well as by constructing bijections for plane trees.
Wagner, Stephan G., Hua Wang.
"On the Parity of the Wiener Index of Trees."
European Journal of Combinatorics, 30 (4): 996-1004.