  We further clarify the relation between detailed-balanced and
complex-balanced equilibria of reversible chemical reaction networks. Our
results hold for arbitrary kinetics and also for boundary equilibria. Detailed
balance, complex balance, "formal balance", and the new notion of "cycle
balance" are all defined in terms of the underlying graph. This fact allows
elementary graph-theoretic (non-algebraic) proofs of a previous result
(detailed balance = complex balance + formal balance), our main result
(detailed balance = complex balance + cycle balance), and a corresponding
result in the setting of continuous-time Markov chains.
