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

The crossing number of a graph is the minimum number of edge crossings over all drawings of in the plane. The main aim of this paper is to give the crossing numbers of the join products of seven graphs on six vertices with paths and cycles on 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. Finally, for at least three and , we also establish the validity of a conjecture introduced by Sta\v s and Valiska concerning the crossings numbers of the join products of the wheels on vertices with the paths on vertices.