On the Jacobian Group for Möbius Ladder and Prism Graphs

Abstract

The notion of the Jacobian group of graph (also known as Picard group, sandpile group, critical group) was independently given by many authors. This is a very important algebraic invariant of a finite graph. In particular, the order of the Jacobian group coincides with the number of spanning trees for a graph. The latter number is known for the simplest families of graphs such as Wheel, Fan, Prism, Ladder and Möbius ladder graphs. At the same time the structure of the Jacobian group is known only in several cases. The aim of this paper is to determine the structure of the Jacobian group of the Möbius ladder and Prism graphs.