Cyclic permutations and crossing numbers of join products of symmetric graph of order six

In the paper, we extend 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 two leaves incident with two opposite vertices of one -cycle, 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 a~calculating the distances between all vertices of the graph. Finally, by adding new edges to the graph , we are able to obtain the crossing number of the join product with the discrete graph for two other graphs. The methods used in the paper are new, and they are based on combinatorial properties of cyclic permutations.