  We propose new Lieb-Robinson bounds (bounds on the speed of propagation of
information in quantum systems) with an explicit dependence on the interaction
strengths of the Hamiltonian. For systems with more than two interactions it is
found that the Lieb-Robinson speed is not always algebraic in the interaction
strengths. We consider Hamiltonians with any finite number of bounded operators
and also a certain class of unbounded operators. We obtain bounds and
propagation speeds for quantum systems on lattices and also general graphs
possessing a kind of homogeneity and isotropy. One area for which this
formalism could be useful is the study of quantum phase transitions which occur
when interactions strengths are varied.
