Müller Stefan
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
J Nonlinear Sci. 2023;33(5):91. doi: 10.1007/s00332-023-09942-w. Epub 2023 Aug 2.
We provide a new decomposition of the Laplacian matrix (for labeled directed graphs with strongly connected components), involving an invertible , 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 (and corresponding polyhedral cones in logarithmic coordinates). As it turns out, all dynamical systems are asymptotically stable that can be embedded in certain . In particular, this holds for complex-balanced mass-action systems, and hence, we also obtain a polyhedral-geometry proof of the classical result.
我们给出了拉普拉斯矩阵的一种新分解（针对具有强连通分量的带标签有向图），涉及一个可逆矩阵、树常数向量以及一个辅助图的关联矩阵，该辅助图表示顶点的一种排序。根据特定的排序，核心矩阵具有额外的性质。我们的结果本质上是图论/代数方面的。作为第一个应用，我们进一步阐明了由化学反应网络产生的（弱可逆）质量作用系统的二项式结构。其次，我们扩展了霍恩和杰克逊关于特殊稳态（复杂平衡平衡点）渐近稳定性的经典结果。在此，图拉普拉斯矩阵的新分解使我们能够考虑正象限中具有给定（以及对数坐标下相应多面锥）的区域。事实证明，所有能够嵌入到某些特定区域的动力系统都是渐近稳定的。特别地，对于复杂平衡质量作用系统也是如此，因此，我们还得到了该经典结果的多面体几何证明。
