Cyclic permutations and crossing numbers of join products of two symmetric graphs of order six

The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product , where the graph consists of one -cycle and of one isolated vertex, and consists on isolated vertices. The proof is done with the~help of software that generates all cyclic permutations for a given number , and creates a~new graph for calculating the distances between all vertices of the graph. Finally, by adding some edges to the graph , we are able to obtain the crossing numbers of the join product with the discrete graph and with the path on vertices for other two graphs.