The crossing numbers of join products of eight graphs of order six with paths and cycles

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. The main aim of this paper is to give the crossing numbers of the join products of eight graphs on six vertices with paths and cycles on $n$ vertices. The proofs are done with the help of several well-known auxiliary statements, the idea of which is extended by a suitable classification of subgraphs that do not cross the edges of the examined graphs.