Let $G_X = (X, \mathcal{E}_X)$ and $G_Y = (Y, \mathcal{E}_Y)$ each be a D778: Graph.
An D548: Ordered pair $G_Z = (Z, \mathcal{E}_Z)$ is a tensor product of $G_X$ and $G_Y$ if and only if

(1)	$Z = X \times Y$ (D326: Cartesian product)
(2)	$\mathcal{E}_Z = \{ \{ (x_1, x_2), (y_1, y_2) \} : \{ x_1, y_1 \} \in \mathcal{E}_X \text{ and } \{ x_2, y_2 \} \in \mathcal{E}_Y \}$