On Wiener index of graph complements

Let G be an (n;m)-graph. We say that G has property () if for every pair of its adjacent vertices x and y, there exists a vertex z, such that z is not adjacent to either x or y. If the graph G has property (), then its complement G is connected, has diameter 2, and its Wiener index is equal to ??n 2
+ m, i.e., the Wiener index is insensitive of any other structural details of the graph G. We characterize numerous classes of graphs possessing property (), among which are trees, regular, and unicyclic graphs.