Graph theoretic approach to the stability analysis of steady state chemical reaction networks

A form of graph theory is developed which makes it possible to write the Routh‐Hurwitz stability conditions for any network of chemical reactions as sums of graphs. These sums, which must all be positive for stability, can contain negative terms only through two mechanisms: first, by having an odd number of certain types of cycles in formally positive graphs, or second, by having an even number of these cycles and at least one cycle in formally negative graphs. The set of graphs in each stability inequality may be represented as a set of points which defines a convex polytope in a higher dimensional space. For large parameter values only the vertices of this polytope affect the stability of the network. For each vertex corresponding to a graph which is a negative term in a stability inequality there is a convex coneshaped contribution to the network's unstable region. For large parameter values, this region is the union of the interiors of these convex cones in parameter space whose boundaries are hyperpl...