  The continuous-time quantum walk is a particle evolving by Schr\"odinger's
equation in discrete space. Encoding the space as a graph of vertices and
edges, the Hamiltonian is proportional to the discrete Laplacian. In some
physical systems, however, the Hamiltonian is proportional to the adjacency
matrix instead. It is well-known that these quantum walks are equivalent when
the graph is regular, i.e., when each vertex has the same number of neighbors.
If the graph is irregular, however, the quantum walks evolve differently. In
this paper, we show that for some irregular graphs, if the particle is
initially localized at a certain vertex, the probability distributions of the
two quantum walks are identical, even though the amplitudes differ. We
analytically prove this for a graph with five vertices and a graph with six
vertices. By simulating the walks on all 1,018,689,568 simple, connected,
irregular graphs with eleven vertices or less, we found sixty-four graphs with
this notion of equivalence. We also give eight infinite families of graphs
supporting these equivalent walks.
