Extremal trees with respect to bi-Wiener index

Abstract

In this paper we introduce and study a new graph-theoretic invariant called the bi-Wiener index. The bi-Wiener index Wb(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 Wb is attained for the star Sn, and the maximum Wb is attained at Pn for even n, or at Pn and Bn(2) for odd n where Bn(2) is a broom with maximum degree 3. We also determine the extremal values of the ratio Wb(Tn)/W(Tn) over all trees on a given number of vertices n. At the end, we indicate some open problems and discuss some possible directions of further research. AMS Subj. Class. (2020): 05C05, 05C09, 05C12, 05C92