Wiener odd and even indices on BC-Trees

Corresponding to the concepts of Wiener index and distance of the vertex, in this paper, we present the concepts of Wiener odd (even) index of G as sum of the distances between all pairs of vertices of G satisfying the distances are all odd (even) and denote them Wodd(G) and Wodd(G) respectively. Based on the concepts of the two indices, we prove theoretically that Wiener odd index is not more than its even index for general BC-Trees. Closed formulae of the two indices are also provided for path BC-tree, star, k-extending star tree and caterpillar BC-tree. Meanwhile, the extreme values of Wodd(T) of n vertices BC-trees are characterized as well.