  We study "positive" graphs that have a nonnegative homomorphism number into
every edge-weighted graph (where the edgeweights may be negative). We
conjecture that all positive graphs can be obtained by taking two copies of an
arbitrary simple graph and gluing them together along an independent set of
nodes. We prove the conjecture for various classes of graphs including all
trees. We prove a number of properties of positive graphs, including the fact
that they have a homomorphic image which has at least half the original number
of nodes but in which every edge has an even number of pre-images. The results,
combined with a computer program, imply that the conjecture is true for all
graphs up to 9 nodes.
