Abstract. Let G be a graph. The common neighborhood graph of G, con(G), has vertex set identical with the vertex set of G, and two vertices are connected by an edge in con(G) if they have a common neighbor in G. If G is not a bipartite graph, then the diameter of con(G) is at most the diameter of G. Nevertheless, there are graphs G for which the Wiener index of con(G) is greater than the wiener index of G. In this paper we present constructions of two infinite classes of unicyclic graphs, which have this property (one with bounded diameter, the other with bounded degree).