On complexity and Jacobian of cone over a graph
| Autor: | Grunwald, L. A., Mednykh, I. A. |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | Grunwald, L. and Mednykh, I. (2021), On the Jacobian group of a cone over a circulant graph, Mathematical notes of NEFU, 28(2), pp. 88-101 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.25587/SVFU.2021.32.84.006 |
| Popis: | 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 $\det(I+L(G)).$ Comment: 14 pages |
| Databáze: | arXiv |
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