The crossing numbers of join products of paths with three graphs of order five

The main aim of this paper is to give the crossing number of the join product \(G^\ast+P_n\) for the disconnected graph \(G^\ast\) of order five consisting of the complete graph \(K_4\) and one isolated vertex, where \(P_n\) is the path on \(n\) vertices. The proofs are done with the help of a lot of well-known exact values for the crossing numbers of the join products of subgraphs of the graph \(G^\ast\) with the paths. Finally, by adding new edges to the graph \(G^\ast\), we are able to obtain the crossing numbers of the join products of two other graphs with the path \(P_n\).