  We show that, in the graph spectrum of the normalized graph Laplacian on
trees, the eigenvalue 1 and eigenvalues near 1 are strongly related to minimum
vertex covers.
  In particular, for the eigenvalue 1, its multiplicity is related to the size
of a minimum vertex cover, and zero entries of its eigenvectors correspond to
vertices in minimum vertex covers; while for eigenvalues near 1, their distance
to 1 can be estimated from minimum vertex covers; and for the largest
eigenvalue smaller than 1, the sign graphs of its eigenvectors take vertices in
a minimum vertex cover as representatives.
