Geodesic
On the Jacobian group of a cone over a circulant graph
Matematičeskie zametki SVFU, Tome 28 (2021) no. 2, pp. 88-101 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: spanning tree, spanning forest, cone over graph, Chebyshev polynomial.
Mots-clés : circulant graph, Laplacian matrix 
@article{SVFU_2021_28_2_a5,
     author = {L. A. Grunwald and I. A. Mednykh},
     title = {On the {Jacobian} group of a cone over a circulant graph},
     journal = {Matemati\v{c}eskie zametki SVFU},
     pages = {88--101},
     year = {2021},
     volume = {28},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SVFU_2021_28_2_a5/}
}
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L. A. Grunwald; I. A. Mednykh. On the Jacobian group of a cone over a circulant graph. Matematičeskie zametki SVFU, Tome 28 (2021) no. 2, pp. 88-101. http://geodesic.mathdoc.fr/item/SVFU_2021_28_2_a5/