On the Jacobian group of a cone over a circulant graph

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.