On the Jacobian group of a cone over a circulant graph

Abstract

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.

Cyclic coverings of graphs. Counting rooted spanning forests and trees, Kirchhoff index, and Jacobians

“…This result was independently obtained by the authors of [16], [47], [32] and others. At the same time, there is very little to say about analytic formulae for the number of spanning forests.…”

“…The following theorem was independently proved by different authors (see, for example, [28], Remark 3,and [32], Theorem 2). Theorem 11.5.…”

“…The following theorem (in an equivalent form) was stated in [60]. The technique employed in its proof was presented in the later work [32].…”

“…Note that coker(I n + L(G)) is a finite Abelian group of order det(I n + L(G)). According to (1), this number coincides with the number of rooted spanning forests in G. Following [32] we call coker(I n + L(G)) the forest group of the graph G and denote it by F (G). Then the statement of Theorem 11.5 can be rephrased as follows:…”

“…The forest group of a circulant graph. For a circulant graph with vertices of even degree the following theorem holds ( [32], Theorem 3).…”