Graph embeddings are useful in bounding the smallest nontrivial eigenvalues
of Laplacian matrices from below. For an n x n Laplacian, these embedding
methods can be characterized as follows: The lower bound is based on a cliq
ue embedding into the underlying graph of the Laplacian. An embedding can b
e represented by a matrix Gamma; the best possible bound based on this embe
dding is n/lambda(max)(Gamma(Gamma)Gamma), where lambda(max) indicates the
largest eigenvalue of the specified matrix. However, the best bounds produc
ed by embedding techniques are not tight; they can be off by a factor propo
rtional to log(2) n for some Laplacians.
We show that this gap is a result of the representation of the embedding: B
y including edge directions in the embedding matrix representation Gamma, i
t is possible to find an embedding such that Gamma(Gamma)Gamma has eigenval
ues that can be put into a one-to-one correspondence with the eigenvalues o
f the Laplacian. Specifically, if lambda is a nonzero eigenvalue of either
matrix, then n/lambda is an eigenvalue of the other. Simple transformations
map the corresponding eigenvectors to each other. The embedding that produ
ces these correspondences has a simple description in electrical terms if t
he underlying graph of the Laplacian is viewed as a resistive circuit. We a
lso show that a similar technique works for star embeddings when the Laplac
ian has a zero Dirichlet boundary condition, though the related eigenvalues
in this case are reciprocals of each other. In the zero Dirichlet boundary
case, the embedding matrix Gamma can be used to construct the inverse of t
he Laplacian. Finally, we connect our results with previous techniques for
producing bounds and provide an example.