Wiener index of uniform hypergraphs induced by trees

The Wiener index W ( G ) of a graph G is defined as the sum of distances between its vertices. A tree T generates r -uniform hypergraph H r , k ( T ) by the following way: hyperedges of cardinality r correspond to edges of the tree and adjacent hyperedges have k vertices in common. A relation between quantities W ( T ) and W ( H r , k ( T ) ) is established.