Spectral characterization of the uniform theta graph Θ(t, 2) and classification of 6-periodic Grover walks

This paper investigates the relationship between the structure of nonregular graphs and the periodicity of Grover walks—a type of discrete-time quantum walk. A graph is considered periodic if its time evolution matrix, after a certain number of steps, returns to the identity matrix. The author seeks to classify all connected graphs that induce a 6-periodic Grover walk. To achieve this, the paper utilizes the spectral mapping theorem, which links the eigenvalues of the Grover walk's time evolution matrix to those of the graph's normalized adjacency matrix. The author also provides a direct spectral characterization of the uniform theta graph Θ(t, 2) using its normalized adjacency spectrum.

The paper establishes three primary results. First, it identifies two families of nonregular graphs that induce periodic Grover walks: the Dutch windmill graph D(t)n, which is 2n-periodic, and the uniform theta graph Θ(t, n), which is (2n + 2)-periodic. Second, the author provides a complete classification of connected 6-periodic graphs, proving that they are exclusively the Dutch windmill graphs D(t)3 (for t ≥ 2) and the uniform theta graphs Θ(t, 2) (for t ≥ 1). Third, the paper offers a spectral characterization of the uniform theta graph Θ(t, 2), demonstrating that it is uniquely determined by its normalized adjacency spectrum.

Understanding the periodicity of quantum walks is essential for applications in quantum information, including quantum cryptography and state transfer protocols. While much of the existing literature focuses on regular graphs, this work extends the analysis to nonregular structures. By classifying 6-periodic graphs, the paper bridges the gap between spectral graph theory and quantum walk dynamics, providing a clear structural characterization that can be used to identify graphs with specific quantum properties.