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In this paper we study a class of discrete quantum walks, known as bipartitewalks. These include the well-known Grover's walks. Any discrete quantum walkis given by the powers of a unitary matrix $U$ indexed by arcs or edges of theunderlying 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 whenthe 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 whosesquares are algebraic integers with degree at most two, we characterizeperiodicity of the bipartite walk over $G$ in terms of its spectrum. We applyperiodicity results of bipartite walks to get a characterization of periodicityof Grover's walk on regular graphs.

Periodicity of bipartite walk on biregular graphs with conditional spectra