TheWiener index W(G) of a graphGis a distance-based topological index
defined as the sum of distances between all pairs of vertices in G. It is shown that for
λ = 2 there is an infinite family of planar bipartite chemical graphs G of girth 4
with the cyclomatic number λ, but their line graphs are not chemical graphs, and
for λ  2 there are two infinite families of planar nonbipartite graphs G of girth 3
with the cyclomatic number λ; the three classes of graphs have the property W(G) =
W(L(G)), where L(G) is the line graph of G.