  The Jacobian matrix of a dynamic system and its principal minors play a
prominent role in the study of qualitative dynamics and bifurcation analysis.
When interpreting the Jacobian as an adjacency matrix of an interaction graph,
its principal minors correspond to sets of disjoint cycles in this graph and
conditions for various dynamic behaviors can be inferred from its cycle
structure. For chemical reaction systems, more fine-grained analyses are
possible by studying a bipartite species-reaction graph. Several results on
injectivity, multistationarity, and bifurcations of a chemical reaction system
have been derived by using various definitions of such bipartite graph. Here,
we present a new definition of the species-reaction graph that more directly
connects the cycle structure with determinant expansion terms, principal
minors, and the coefficients of the characteristic polynomial and encompasses
previous graph constructions as special cases. This graph has a direct relation
to the interaction graph, and properties of cycles and sub-graphs can be
translated in both directions. A simple equivalence relation enables to
decompose determinant expansions more directly and allows simpler and more
direct proofs of previous results.
