Spectral characterization of the uniform theta graph $\mathrm{\Theta }(t,2)$ and classification of 6-periodic Grover walks

We characterize the uniform theta graph $\mathrm{\Theta }(t,2)$ by the spectrum of its normalized adjacency matrix, or equivalently, by the spectrum of its normalized Laplacian matrix. We also investigate the periodicity of Grover walks on nonregular graphs, which is closely related to the eigenvalues of the normalized adjacency matrix and those of the time evolution matrix of the Grover walk. We show that the Dutch windmill graph $D_n^{(t)}$ is $2n$-periodic and that the uniform theta graph $\mathrm{\Theta }(t,n)$ is $(2n+2)$-periodic. Furthermore, we completely determine the connected $6$-periodic graphs and prove that they are precisely $D_3^{(t)}$ with $t\ge 2$ and $\mathrm{\Theta }(t,2)$ with $t\ge 1$.