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We consider the family of graphs whose vertex set is $\mathbb{Z}^n$ where twovertices are connected by an edge when their $\ell_\infty$-distance is 1.Towards an edge isoperimetric inequality for this graph, we calculate the edgeboundary of any finite set $S \subset \mathbb{Z}^n$. This boundary calculationleads to a desire to show that a set with optimal edge boundary has no ``gaps''in any direction $\epsilon \in \{-1,0,1\}^n, \epsilon \not=0$. We show that onecan find a set with optimal edge boundary that does not have gaps in anydirection $e_i$ (or $-e_i$) where $e_i$ is the standard basis vector.

Edge Boundaries for a Family of Graphs on $\mathbb{Z}^n$