A new decomposition of the graph Laplacian and the binomial structure of mass-action systems

Abstract

We provide a new decomposition of the Laplacian matrix (for labeled directed graphs with strongly connected components), involving an invertible $\textit{core matrix}$, the vector of tree constants, and the incidence matrix of an auxiliary graph, representing an order on the vertices. Depending on the particular order, the core matrix has additional properties. Our results are graph-theoretic/algebraic in nature. As a first application, we further clarify the binomial structure of (weakly reversible) mass-action systems, arising from chemical reaction networks. Second, we extend a classical result by Horn and Jackson on the asymptotic stability of special steady states (complex-balanced equilibria). Here, the new decomposition of the graph Laplacian allows us to consider regions in the positive orthant with given $\textit{monomial evaluation orders}$ (and corresponding polyhedral cones in logarithmic coordinates). As it turns out, all dynamical systems are asymptotically stable that can be embedded in certain $\textit{binomial differential inclusions}$. In particular, this holds for complex-balanced mass-action systems, and hence we also obtain a polyhedral-geometry proof of the classical result.