  The study of the dynamics of chemical reactions, and in particular phenomena
such as oscillating reactions, has led to the recognition that many dynamical
properties of a chemical reaction can be predicted from graph theoretical
properties of a certain directed graph, called a Chemical Reaction Network
(CRN). In this graph, the edges represent the reactions and the vertices the
reacting combinations of chemical substances.
  In contrast with the classical treatment, in this work, we heavily rely on a
recently developed theory of directed graph Laplacians to simplify the
traditional treatment of the so-called deficiency zero systems of CRN theory.
We show that much of the dynamics of these polynomial systems of differential
equations can be understood by analyzing the directed graph Laplacian
associated with the system. Beside the more concise mathematical treatment,
this leads to considerably stronger results. In particular, (i) we show that
our Laplacian deficiency zero theorem is markedly stronger than the traditional
one and (ii) we derive simple equations for the locus of the equilibria in all
(Laplacian) deficiency zero cases.
  This paper is written in a way to make the material easily accessible to a
mathematical audience. In particular, no knowledge of chemistry or physics is
assumed.
