  This work explores the definiteness of the weighted graph Laplacian matrix
with negative edge weights. The definiteness of the weighted Laplacian is
studied in terms of certain matrices that are related via congruent and
similarity transformations. For a graph with a single negative weight edge, we
show that the weighted Laplacian becomes indefinite if the magnitude of the
negative weight is less than the inverse of the effective resistance between
the two incident nodes. This result is extended to multiple negative weight
edges. The utility of these results are demonstrated in a weighted consensus
network where appropriately placed negative weight edges can induce a
clustering behavior for the protocol.
