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

The crossing number $\mathrm{cr}(G)$ of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. In the paper, the crossing number of the join product $G^\ast + D_n$ for the connected graph $G^\ast$ on six vertices consisting of one path on four vertices $P_4$ and two leaves adjacent with the same outer vertex of the path $P_4$ is given, where $D_n$ consists of $n$ isolated vertices. Finally, by adding some edges to the graph $G^\ast$, we obtain the crossing numbers of the join products of other four graphs with $D_n$.