The hyper-Wiener index of graphs with given bipartition

Abstract

Let G be a simple connected graph. The Wiener index W(G) is the sum of all distances between vertices of G, whereas the hyper-Wiener index WW(G) is defined as WW(G) = 1/2 Σ[u,v]⊆v(G)(d(u,v)+d²(u,v)), with the summation going over all pairs of vertices in G. In this paper, we obtain the sharp upper or lower bounds for the hyper-Wiener indices among trees or bipartite unicyclic graphs with given bipartition, we also characterize the corresponding extremal graphs.