The hyper-Wiener index of graphs with a given chromatic (clique) number

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 hyperWiener index WW(G) is defined as WW(G) = d ²(u,v)), with the summation going over all pairs of vertices in G. In this paper, we determine the extremal graphs of order n and chromatic (clique) number k with maximal and minimal hyper-Wiener index.