  We study cospectral vertices on finite graphs in relation to the echolocation
problem on Riemannian manifolds. First, We prove a computationally simple
criterion to determine whether two vertices are cospectral. Then, we use this
criterion in conjunction with a computer search to find minimal examples of
various types of graphs on which cospectral but non-similar vertices exist,
including minimal walk-regular non-vertex-transitive graphs, which turn out to
be non-planar. Moreover, as our main result, we classify all finite 3-connected
walk-regular planar graphs, proving that such graphs must be vertex-transitive.
