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 \(n\ge 4\), 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.