On a time-frequency approach to translation on finite graphs

The authors of [1] have used spectral graph theory to define a Fourier transform on finite graphs. With this definition, one can use elementary properties of classical time-frequency analysis to define time-frequency operations on graphs including convolution, modulation, and translation. Many of these graph operators have properties that match our intuition in Euclidean space. The exception lies with the translation operator. In particular, translation does not form a group, i.e., T_iT_j ≠ T_i+j. We prove that graphs whose translation operators exhibit semigroup behavior are those whose eigenvectors of the Laplacian form a Hadamard matrix.