Jacobian groups of circulant graphs

The notion of the Jacobian group of a graph, which is also known as the Picard group, the critical group, and the dollar or sandpile group, was independently introduced by many authors. We define Jacobian of a graph as the maximal Abelian group generated by the flows obeying two Kirchhoffs laws. This notion arise as a discrete version of the Jacobian in the classical theory of Riemann surfaces. It also admits a natural interpretation in various areas of physics, coding theory, and financial mathematics. The Jacobian group is an important algebraic invariant of a finite graph. In particular, it's order coincides with the number of spanning trees of the graph, which is well known for some simplest graphs, such as the wheel, fan, prism, ladder, and Mobius ladder. At the same time, the structure of the Jacobian is known only in some particular cases. The class of circulant graphs is fairly large and includes the cyclic graphs, complete graphs, Mobius ladder, antiprisms, and other graphs. The purpose of this report is to determine the structure of the Jacobian for circulant graphs, the generalized Petersen graph, I-, Y-, H-graphs and some others. We also present new formulas for the number of spanning trees and investigate arithmetical properties of these numbers.