TY - JOUR
AU - Liliya Grunwald
AU - Ilya Mednykh
PY - 2021/07/20
Y2 - 2024/09/13
TI - On the Jacobian group of a cone over a circulant graph
JF - Mathematical notes of NEFU
JA - Math. notes NEFU
VL - 28
IS - 2
SE - Mathematics
DO - https://doi.org/10.25587/SVFU.2021.32.84.006
UR - https://mzsvfu.ru/index.php/mz/article/view/on-the-jacobian-group-of-a-cone-over-a-circulant-graph
AB - For any given graph G, consider the graph Ĝ 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 Ĝ coincides with the number of rooted spanning forests in G and the Jacobian of Ĝ 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 Ĝ as det(I + L(G)).As an application, we establish general structural theorems for the Jacobian of Ĝ in the case when G is a circulant graph or cobordism of two circulant graphs.
ER -