On Wiener Index of Common Neighborhood Graphs

If G is a simple graph, then con(G), the common neighborhood graph of G, has the same vertex set as G, and two vertices of con(G) are adjacent if they have a common neighbor in G. We show that for any bipartite graph G the Wiener index (i.e., sum of distances between all pairs of vertices) of con(G) is always smaller than the Wiener index of G. For general graphs, however, the Wiener index of common neighbor graphs can be greater. This fact is surprising, since we also show that the diameter of con(G) is at most the diameter of G. We present constructions of two infinite classes of graphs, chemical and unicyclic :graphs, which have this property.