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

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.

Periodicity of bipartite walk on biregular graphs with conditional spectra

“…Once we have the formula, by the discussion of previous section, we can use Theorem 4.1 to give a characterization of periodicity of Grover's walk on regular graphs whose eigenvalues λ satisfies that λ 2 is a algebraic integer with degree at most two. This characterization extends the result in [15].…”

“…Note that when we talk about biregular graph in the context of bipartite walk, we assume that all the vertices in the same color class have the same degree. We show that if a biregular graph G has eigenvalues whose squares are algebraic integers with degree at most two, there is a characterization of periodicity of bipartite walks in terms of spectrum of G. As we stated before, this result extends the result proved by Kubota in [15].…”

“…Results in this paper extend the periodicity results of Grover's walks proved by Kubota in [15]. More specifically, results in this paper remove the constraint on the number of distinct eigenvalues and we do not require the underlying regular graph to be bipartite.…”

“…We only requires that the underlying graph is regular and it has eigenvalues whose squares are algebraic integers with degree at most two. There will be examples of periodic graphs shown in the paper that are not explained by the results in [15].…”

“…Recently in [15], Kubota studies the periodicity of Grover's walk on regular bipartite graphs with at most five distinct eigenvalues. Since the number of distinct eigenvalues of graphs are at most five, it implies all the eigenvalues of graphs of interest in [15] are algebraic integers with degree at most two. Later in the paper, we show that Grover's walk is a special case of bipartite walk.…”