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 edge crossings over all drawings in plane. main aim this paper to give numbers join products eight graphs on six vertices with paths and cycles $n$ vertices. proofs are done help several well-known auxiliary statements, idea which extended by suitable classification subgraphs that do not cross edges examined graphs.