the crossing numbers of join products of $k_4cup k_1$ with cycles

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. in the paper, we extend known results concerning crossing numbers of join products of two small graphs with cycles. the crossing number of the join product $g^ast + c_n$ for the disconnected graph $g^ast$ consisting of the complete graph $k_{4}$ and one isolated vertex is given, where $c_n$ is the cycle on $n$ vertices. the proof of the main result is done with the help of lemma whose proof is based on a special redrawing technique. up to now, the crossing numbers of $g + c_n$ were done only for a few disconnected graphs $g$. finally, by adding new edge to the graph $g^ast$, we are able to obtain the crossing number of $g_1+c_n$ for one other graph $g_1$ of order five.