Reverse Wiener indices of graphs of exactly two cycles

Abstract

The reverse Wiener index of a connected graph G with n vertices is defined as A(G) = 1/2n(n-1)d-W(G) where d and W(G)are respectively the diameter and Wiener index of G. We determine the n-vertex connected graph(s) of exactly two cycles of a vertex in common with the k-th greatest reverse Wiener indices for all k up to three if n = 7, four if n = 8, [√n-7/2]+1 if n ≥ 9, and the n-vertex connected graph(s) of exactly two vertex-disjoint cycles with the greatest reverse Wiener index. The n-vertex connected graphs with exactly two cycles with the greatest reverse Wiener index are determined for n≥7.