The main aim of the paper is to give the crossing number of the join product $G+D_n$ for the disconnected graph G of order five consisting of one isolated vertex and of one vertex incident with some vertex of the three-cycle, and $D_n$ consists of n isolated vertices. In the proofs, the idea of the new representation of the minimum numbers of crossings between two different subgraphs that do not cross the edges of the graph G by the graph of configurations ${\mathcal{G}}_D$ in the considered drawing D of $G+D_n$ will be used. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph $D_n$ and with the path $P_n$ on n vertices for three other graphs.

The crossing number $\mathrm{cr}\left(G\right)$ of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product $K_{2,3}+C_n$ for the complete bipartite graph $K_{2,3}$ , where $C_n$ is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph $K_{2,3}$ , we also offer the crossing number of the join product of one other graph with the cycle $C_n$ .

The crossing number \(\mathrm{cr}(G)\) of a graph \(G\) is the minimum number of edge crossings over all drawings of \(G\) in the plane. The main aim of the paper is to give the crossing number of the join product \(W_4+P_n\) and \(W_4+C_n\) for the wheel \(W_4\) on five vertices, where \(P_n\) and \(C_n\) are the path and the cycle on \(n\) vertices, respectively. Yue et al. conjectured that the crossing number of \(W_m+C_n\) is equal to \(Z(m+1)Z(n)+(Z(m)-1) \big \lfloor \frac{n}{2} \big \rfloor + n+ \big\lceil\frac{m}{2}\big\rceil +2\), for all \(m,n \geq 3\), and where the Zarankiewicz's number \(Z(n)=\big \lfloor \frac{n}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor\) is defined for \(n\geq 1\). Recently, this conjecture was proved for \(W_3+C_n\) by Klešč. We establish the validity of this conjecture for \(W_4+C_n\) and we also offer a new conjecture for the crossing number of the join product \(W_m+P_n\) for \(m\geq 3\) and \(n\geq 2\).

The main aim of the paper is to establish the crossing numbers of the join products of the paths and the cycles on n vertices with a connected graph on five vertices isomorphic to the graph $K_{1,1,3}\backslash e$ obtained by removing one edge e incident with some vertex of order two from the complete tripartite graph $K_{1,1,3}$. The proofs are done with the help of well-known exact values for the crossing numbers of the join products of subgraphs of the considered graph with paths and cycles. Finally, by adding some edges to the graph under consideration, we obtain the crossing numbers of the join products of other graphs with the paths and the cycles on n vertices.

We extend known results concerning crossing numbers by giving the crossing number of the join product $G+D_{n}$ , where the connected graph $G$ consists of one $4$ -cycle and of two leaves incident with the same vertex of the $4$ -cycle, and $D_{n}$ consists of $n$ isolated vertices. The proofs are done with the help of software that generates all cyclic permutations for a given number $k$ and creates a graph for calculating the distances between all $(k-1)!$ vertices of the graph.