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A \v{C}ech complex of a finite simple graph $G$ is a nerve complex of ballsin the graph, with one ball centered at each vertex. More precisely, let the\v{C}ech complex $\mathcal{N}(G,r)$ be the nerve of all closed balls of radius$\frac{r}{2}$ centered at vertices of $G$, where these balls are drawn in thegeometric realization of the graph $G$ (equipped with the shortest pathmetric). The simplicial complex $\mathcal{N}(G,r)$ is equal to the graph $G$when $r=1$, and homotopy equivalent to the graph $G$ when $r$ is smaller thanhalf the length of the shortest loop in $G$. For higher values of $r$, thetopology of $\mathcal{N}(G,r)$ is not well-understood. We consider the$n$-dimensional hypercube graphs $\mathbb{I}_n$ with $2^n$ vertices. Our mainresults are as follows. First, when $r=2$, we show that the \v{C}ech complex$\mathcal{N}(\mathbb{I}_n,2)$ is homotopy equivalent to a wedge of 2-spheresfor all $n\ge 1$, and we count the number of 2-spheres appearing in this wedgesum. Second, when $r=3$, we show that $\mathcal{N}(\mathbb{I}_n,3)$ is homotopyequivalent to a simplicial complex of dimension at most 4, and that for $n\ge4$ the reduced homology of $\mathcal{N}(\mathbb{I}_n, 3)$ is nonzero indimensions 3 and 4, and zero in all other dimensions. Finally, we show that forall $n\ge 1$ and $r\ge 0$, the inclusion $\mathcal{N}(\mathbb{I}_n,r)\hookrightarrow \mathcal{N}(\mathbb{I}_n, r+2)$ is null-homotopic, providinga bound on the length of bars in the persistent homology of \v{C}ech complexesof hypercube graphs.

\v{C}ech complexes of hypercube graphs