关于图上锥的复杂性和Jacobian
On complexity and Jacobian of cone over a graph
摘要：
对于任意给定的图$G$，考虑图$\widetilde{G}$是图$G.$上的锥。本文研究了锥的两个重要不变量。即，复杂度（生成树的数量）和图的雅可比矩阵。证明了图$\widetilde{G}$的复杂度与图$G$的根生成森林数一致，并且$\widetilde{G}$的Jacobian同构于算子$I+L(G),$的上核，其中$L(G)$是$G$的Laplacian，$I$是单位矩阵.因此，可以将$\widetilde{G}$的复杂度计算为$\det(I+L(G)).$
英文摘要：
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 $\det(I+L(G)).$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
