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 W_odd(G) and W_odd(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 W_odd(T) of n vertices BC-trees are characterized as well.