  Let $\Delta+V$ be the discrete Schr\"odinger operator, where $\Delta$ is the
discrete Laplacian on $\mathbb{Z}^d$ and the potential $V:\mathbb{Z}^d\to
\mathbb{C}$ is $\Gamma$-periodic with $\Gamma=q_1\mathbb{Z}\oplus q_2
\mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$. In this study, we establish a
comprehensive characterization of the complex-valued $\Gamma$-periodic
functions such that the Bloch variety of $\Delta+V$ contains the graph of an
entire function, in particular, we show that there are exactly $q_1q_2\cdots
q_d$ such functions (up to the Floquet isospectrality and the translation).
Moreover, by applying this understanding to real-valued functions $V$, we
confirm the conjecture concerning the geometric version of Borg's theorem in
arbitrary dimensions.
