  The graph Laplacian operator is widely studied in spectral graph theory
largely due to its importance in modern data analysis. Recently, the Fourier
transform and other time-frequency operators have been defined on graphs using
Laplacian eigenvalues and eigenvectors. We extend these results and prove that
the translation operator to the $i$'th node is invertible if and only if all
eigenvectors are nonzero on the $i$'th node. Because of this dependency on the
support of eigenvectors we study the characteristic set of Laplacian
eigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish
on large neighborhoods and then explicitly construct a family of non-planar
graphs that do exhibit this property.
