Given a set of points $X$, an empty-region graph is a graph in which $p$, $q \in X$ are neighbors if some region defined by $(p, q)$ does not contain any point of $X$. We provide expected analyses of the degree of a point and the possibility of having far neighbors in such a graph when $X$ is a planar Poisson point process. Namely the expected degree of a point in the empty axis-alignedellipse graph for a Poisson point process of intensity $\lambda$ in the unit square is $\Theta(\ln\lambda)$. It is $\Theta(\ln\beta)$ if the ellipses are constrained to have an aspect ratio between 1 and $\beta>1$, and $\Theta(\beta)$ when the aspect ratio is constrained but ellipses are not axis-aligned.