The hyper-Wiener index of unicyclic graphs

Abstract

Let G be a connected simple graph. The Wiener index W(G) is the sum of all distances between vertices of G, whereas the hyperWiener index WW(G) is defined as WW(G) = 1/2 Σ_{u,v}∈V(G)(d(u,v) +d²(u, v)). In this paper, we determine the extremal unicyclic graphs with given girth having maximal and minimal hyper-Wiener index.