Wiener-Number-Related Sequences

Recently various generalizations or extensions of the now “classical” Wiener number1 have become of interest, both as regards new individual numbers2-7 and as regards sequences of numbers.8-12 Of particular interest here are those numbers of Tratch et al.2 and of Randić,3 their manner of interrelation, their formulas for trees as extended to general cycle-containing graphs, and their extension to a series of numbers, which turn out to be related to the moments of the distribution of distances. The numbers of interest for the case of a tree graph T (with N vertices) may be obtained via the following algorithm. 0. For each pair of vertices i, j of T identify the unique path π ≡ π(i, j) of length dij between i and j. 1. Remove from T all edges that belong to π and delete all vertices of π except i and j. Delete any side branches attached to these internal vertices of π. The remaining subgraph Tπ is disconnected, possessing two components, with vertex counts of (say) aπ and bπ. 2. Form the sums (over the N(N 1)/2 pairs of vertices {i, j}).