  We develop the theory of fractional revival in the quantum walk on a graph
using its Laplacian matrix as the Hamiltonian. We first give a spectral
characterization of Laplacian fractional revival, which leads to a polynomial
time algorithm to check this phenomenon and find the earliest time when it
occurs. We then apply the characterization theorem to special families of
graphs. In particular, we show that no tree admits Laplacian fractional revival
except for the paths on two and three vertices, and the only graphs on a prime
number of vertices that admit Laplacian fractional revival are double cones.
Finally, we construct, through Cartesian products and joins, several infinite
families of graphs that admit Laplacian fractional revival; some of these
graphs exhibit polygamous fractional revival.
