  We study scattering quantum walks on highly symmetric graphs and use the
walks to solve search problems on these graphs. The particle making the walk
resides on the edges of the graph, and at each time step scatters at the
vertices. All of the vertices have the same scattering properties except for a
subset of special vertices. The object of the search is to find a special
vertex. A quantum circuit implementation of these walks is presented in which
the set of special vertices is specified by a quantum oracle. We consider the
complete graph, a complete bipartite graph, and an $M$-partite graph. In all
cases, the dimension of the Hilbert space in which the time evolution of the
walk takes place is small (between three and six), so the walks can be
completely analyzed analytically. Such dimensional reduction is due to the fact
that these graphs have large automorphism groups. We find the usual quadratic
quantum speedups in all cases considered.
