Graph Jacobians and homology groups of branched knot covers

The notion of a graph Jacobian group, also called the Picard group, critical group, dollar group, or sand group, has been independently introduced by many authors. This is the maximal Abelian group generated by flows on the graph that satisfy the first and second Kirchhoff laws. This concept also appears as a discrete version of the Jacobian in the classical theory of Riemann surfaces. It allows a natural interpretation in various branches of physics, coding theory and financial mathematics. The Jacobian group is an important algebraic invariant of a finite graph. In particular, its order coincides with the number of spanning trees of the graph.
The purpose of this lecture is to determine the structure of the Jacobian for circulant graphs. For the simplest circulant graphs, the Jacobian group will be found explicitly, while in the general case a convenient method for calculating it will be proposed.
A parallel will be noted between the results describing the homology of branched cyclic coverings over knots and the theory of Jacobians of cyclic coverings over graphs.