I am defining a graph operation $\pmb\nabla$ in the following.
The notation $\pmb\nabla$ is a modified form of graph join operation, and defined as follows. Consider two graphs $G$ and $H$, then $G\pmb\nabla H=\langle V(G)\cup V(H), E(G)\cup E(H)\cup \lbrace uv: u\in V(G), v\in V(H);~u\notin V(H), v\notin V(G) \rbrace\rangle.$
This definition is closely related to graph join operation with the difference being a restriction that $u\notin V(H), v\notin V(G)$. Note that this operation is closed and associative.
Does this operation have a name? Any important algebraic properties of this operation?