The Anderson model is actually a random Schrödinger operator (RSO) on the d-dimensional integer lattice Z d , which belongs to the category of infinite graphs with bounded degrees. In this paper, we would like to introduce and investigate a RSO on an infinite graph with unbounded degrees, which we call the infinite-dimensional hypercube (IDH). We first give the definition of the IDH and examine its algebraic and topological properties. We then construct the Laplacian on the IDH and explicitly obtain its spectrum. With the above Laplacian as the free Hamiltonian, we construct a RSO on the IDH and pursue its elementary properties. Finally, we investigate the ergodicity and density of states of the RSO and obtain several results. Some other related results are also proven. [ABSTRACT FROM AUTHOR]