  The quantum walk dynamics obey the laws of quantum mechanics with an extra
locality constraint, which demands that the evolution operator is local in the
sense that the walker must visit the neighboring locations before endeavoring
to distant places. Usually, the Hamiltonian is obtained from either the
adjacency or the laplacian matrix of the graph and the walker hops from
vertices to neighboring vertices. In this work, we define a version of the
continuous-time quantum walk that allows the walker to hop from vertices to
edges and vice versa. As an application, we analyze the spatial search
algorithm on the complete bipartite graph by modifying the new version of the
Hamiltonian with an extra term that depends on the location of the marked
vertex or marked edge, similar to what is done in the standard continuous-time
quantum walk model. We show that the optimal running time to find either a
vertex or an edge is $O(\sqrt{N_e})$ with success probability $1-o(1)$, where
$N_e$ is the number of edges of the complete bipartite graph.
