Properties of the hyper-wiener index as a local function

Hyper-Wiener index was introduced as one of the main generalizations of the well known Wiener index. Through the years properties of the Wiener index have been extensively studied in both Mathematics and Chemistry. The hyper-Wiener index, although received much attention, is far from being thoroughly examined due to its complex definition. We consider the local version of the hyper-Wiener index (WW(G)), defined (equation presented) for a vertex v in a graph G, in trees. For established results on the Wiener index (W(.)), we present analogous studies on WW(.). In addition to interesting observations, some conjectures and questions are also proposed.