Periodicity of Grover walks on bipartite regular graphs with at most five distinct eigenvalues

TL;DR

This paper characterizes bipartite regular graphs with specific eigenvalue properties that lead to periodic Grover walks, identifying only certain graphs like C6 and classifying eigenvalues for graphs with five eigenvalues.

Contribution

It provides a complete characterization of bipartite regular graphs with four and five eigenvalues that induce periodic Grover walks, including enumeration of feasible spectra.

Findings

C6 is the only connected bipartite regular graph with four eigenvalues inducing periodic Grover walks.

Identifies three possible second largest eigenvalues for bipartite regular graphs with five eigenvalues.

Enumerates feasible spectra for such graphs using walk-regularity.

Abstract

We determine connected bipartite regular graphs with four distinct adjacency eigenvalues that induce periodic Grover walks, and show that it is only $C_6$. We also show that there are only three kinds of the second largest eigenvalues of bipartite regular periodic graphs with five distinct eigenvalues. Using walk-regularity, we enumerate feasible spectra for such graphs.