On the crossing numbers of the join products of five graphs on six vertices with discrete graphs

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 .