Fractional revival of threshold graphs under Laplacian dynamics

We consider Laplacian fractional revival between two vertices of a graph X. Assume that it occurs at time τ between vertices 1 and 2. We prove that for the spectral decomposition L=∑r=0qθrEr L = \sum\nolimits_{r = 0}^q {{\theta _r}{E_r}} of the Laplacian matrix L of X, for each r = 0, 1, . . . , q, either Ere1 = Ere2, or Ere1 = −Ere2, depending on whether eiτθr equals to 1 or not. That is to say, vertices 1 and 2 are strongly cospectral with respect to L. We give a characterization of the parameters of threshold graphs that allow for Laplacian fractional revival between two vertices; those graphs can be used to generate more graphs with Laplacian fractional revival. We also characterize threshold graphs that admit Laplacian fractional revival within a subset of more than two vertices. Throughout we rely on techniques from spectral graph theory.