  In this paper we study a class of discrete quantum walks, known as bipartite
walks. These include the well-known Grover's walks. Any discrete quantum walk
is given by the powers of a unitary matrix $U$ indexed by arcs or edges of the
underlying graph. The walk is periodic if $U^k=I$ for some positive integer
$k$. Kubota has given a characterization of periodicity of Grover's walk when
the walk is defined on a regular bipartite graph with at most five eigenvalues.
We extend Kubota's results--if a biregular graph $G$ has eigenvalues whose
squares are algebraic integers with degree at most two, we characterize
periodicity of the bipartite walk over $G$ in terms of its spectrum. We apply
periodicity results of bipartite walks to get a characterization of periodicity
of Grover's walk on regular graphs.
