The crossing number of a graph
is the minimum number of edge crossings over all drawings of
in the plane. In the paper, the crossing number of the join product
for the connected graph
on six vertices consisting of one path on four vertices
and two leaves adjacent with the same outer vertex of the path
is given, where
consists of
isolated vertices. Finally, by adding some edges to the graph
, we obtain the crossing numbers of the join products of other four graphs with
.