On the structure of the Jacobian group for circulant graphs

Abstract

The Jacobian of a graph is defined as the maximal Abelian group generated by flows obeying two Kirchhoff’s laws. This notion, also known as the Picard group, sandpile group, or critical group, has been extensively studied by many authors in the past decade. This is an important algebraic invariant of a finite graph. At the same time, the structure of the Jacobian is known only in particular cases. The paper is devoted to the study of the structure of the Jacobian group for circulant graphs. For the simplest graphs in this family, the Jacobian group is explicitly described, and in the general case, and effective algorithm for calculating it is proposed.