论文标题
关于锥体的复杂性和锥形
On complexity and Jacobian of cone over a graph
论文作者
论文摘要
对于任何给定的图形$ g $，请考虑图$ \ widetilde {g} $，这是图$G。$。在本文中，我们研究了这种锥体的两个重要不变性。也就是说，复杂性（跨越树的数量）和图的雅各布式。我们证明，图$ \ widetilde {g} $的复杂性重合了Graph $ g $中的生根跨越森林的数量，而$ \ widetilde {g} $的jacobian是对操作员的cokernel的$ i+i+l（g），$ l（g）$ l（g）$ laplac is laplac is $ g iS $ g iS $ g iS $ i rix the $ l（g）的cokernel。结果，可以计算$ \ widetilde {g} $的复杂性为$ \ det（i+l（g））。$
For any given graph $G$ consider a graph $\widetilde{G}$ which is a cone over graph $G.$ In this paper, we study two important invariants of such a cone. Namely, complexity (the number of spanning trees) and the Jacobian of a graph. We prove that complexity of graph $\widetilde{G}$ coincides the number of rooted spanning forests in graph $G$ and the Jacobian of $\widetilde{G}$ is isomorphic to cokernel of the operator $I+L(G),$ where $L(G)$ is Laplacian of $G$ and $I$ is the identity matrix. As a consequence, one can calculate the complexity of $\widetilde{G}$ as $\det(I+L(G)).$
