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Эта публикация цитируется в 1 научной статье (всего в 1 статье)
Математика
On the Jacobian group of a cone over a circulant graph
L. A. Grunwaldab, I. A. Mednykhab a Sobolev Institute of Mathematics, 4 Koptyug Avenue, Novosibirsk 630090, Russia
b Novosibirsk State University, 1 Pirogov Street, Novosibirsk 630090, Russia
Аннотация:
For any given graph $G$, consider the graph $\hat{G}$ which is a cone over $G$. We study two important invariants of such a cone, namely, the complexity (the number of spanning trees) and the Jacobian of the graph. We prove that complexity of graph $\hat{G}$ coincides with the number of rooted spanning forests in $G$ and the Jacobian of $\hat{G}$ is isomorphic to the cokernel of the operator $I+L(G)$, where $L(G)$ is the Laplacian of $G$ and $I$ is the identity matrix. As a consequence, one can calculate the complexity of $\hat{G}$ as $\det(I+L(G))$.
As an application, we establish general structural theorems for the Jacobian of $\hat{G}$ in the case when $G$ is a circulant graph or cobordism of two circulant graphs.
Ключевые слова:
spanning tree, spanning forest, circulant graph, Laplacian matrix, cone over graph, Chebyshev polynomial.
Поступила в редакцию: 15.02.2021 Исправленный вариант: 12.03.2021 Принята в печать: 26.05.2021
Образец цитирования:
L. A. Grunwald, I. A. Mednykh, “On the Jacobian group of a cone over a circulant graph”, Математические заметки СВФУ, 28:2 (2021), 88–101
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/svfu319 https://www.mathnet.ru/rus/svfu/v28/i2/p88
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Страница аннотации: | 81 | PDF полного текста: | 49 |
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