Abstract: The crossing numbers of a graph is a vital parameter. Garey and Johnson showed that the problem of determining the crossing numbers of an arbitrary graph is NP-complete. Because of its difficultly, the classes of graphs whose crossing numbers have been determined are very scarce at present. Based on the result of the crossing number of complete bipartite graph cr(K₆,n)=Z(6,n) given by Kleitman, in this paper, for a 6-vertex graph H, it is shown that the crossing numbers of its join with n isolated vertices as well as the path P_n on n vertices and with the cycle C_n are $c r\left(H+n K_{1}\right)=Z(6, n)+2\left\lfloor\frac{n}{2}\right\rfloor$, $c r\left(H+P_{n}\right)=Z(6, n)+2\left\lfloor\frac{n}{2}\right\rfloor$, and $c r\left(H+C_{n}\right)=Z(6, n)+2\left\lfloor\frac{n}{2}\right\rfloor$.

Key words: drawing, crossing number, joint graph, path, cycle