The join $\displaystyle G=G_1+ G_2$ of graphs $\displaystyle G_1$ and $\displaystyle G_2$ with disjoint point sets $\displaystyle V_1$ and $\displaystyle V_2$ and edge sets $\displaystyle X_1$ and $\displaystyle X_2$ is the graph union $\displaystyle G_1 \cup G_2$ together with all the edges joining $\displaystyle V_1$ and $\displaystyle V_2$.

Suppose $\displaystyle G_1=G_2=$ the complete $\displaystyle r$-partite graph $\displaystyle K_{n, \ldots,n}$ then what is the graph $\displaystyle G=G_1+ G_2 ?$

i.e, can we write $\displaystyle G$ by some well known graphs and their operations ?.