Extremal Trees with Respect to Bi-Wiener Index

In this paper we introduce and study a new graph-theoretic invariant called the bi-Wiener index. The bi-Wiener index W b ( G ) of a bipartite graph G is defined as the sum of all (shortest-path) distances between two vertices from different parts of the bipartition of the vertex set of G . We start with providing a motivation connected with the potential uses of the new invariant in the QSAR/QSPR studies. Then we study its behavior for trees. We prove that, among all trees of order n ≥ 4 , the minimum value of W b is attained for the star S n , and the maximum W b is attained at path P n for even n , or at path P n and B n ( 2 ) for odd n where B n ( 2 ) is a broom with maximum degree 3. We also determine the extremal values of the ratio W b ( T n ) / W ( T n ) over all trees of order n . At the end, we indicate some open problems and discuss some possible directions of further research.