  A quantum particle evolving by Schr\"odinger's equation contains, from the
kinetic energy of the particle, a term in its Hamiltonian proportional to
Laplace's operator. In discrete space, this is replaced by the discrete or
graph Laplacian, which gives rise to a continuous-time quantum walk. Besides
this natural definition, some quantum walk algorithms instead use the adjacency
matrix to effect the walk. While this is equivalent to the Laplacian for
regular graphs, it is different for non-regular graphs, and is thus an
inequivalent quantum walk. We algorithmically explore this distinction by
analyzing search on the complete bipartite graph with multiple marked vertices,
using both the Laplacian and adjacency matrix. The two walks differ
qualitatively and quantitatively in their required jumping rate, runtime,
sampling of marked vertices, and in what constitutes a natural initial state.
Thus the choice of the Laplacian or adjacency matrix to effect the walk has
important algorithmic consequences.
