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On the Jacobian Group for Möbius Ladder and Prism Graphs

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On the Jacobian Group for Möbius Ladder and Prism Graphs Fifteenth International Conference on Geometry, Integrability and Quantization June 7–12, 2013, Varna, Bulgaria Ivaïlo M. Mladenov, Andrei Ludu and Akira Yoshioka, Editors Avangard Prima, Sofia 2014, pp 117–126 doi: 10.7546/giq-15-2014-117-126 ON THE JACOBIAN GROUP FOR MÖBIUS LADDER AND PRISM GRAPHS y MADINA DERYAGINA and ILIA MEDNYKH Mechanics and Mathematics Department, Novosibirsk State University 630090 Novosibirsk, Russia ySobolev Institute of Mathematics, Novosibirsk State University 630090 Novosibirsk, Russia Abstract. The notion of the Jacobian group of graph (also known as Picard group, sandpile group, critical group) was independently given by many au- thors. This is a very important algebraic invariant of a finite graph. In partic- ular, 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. 1. Introduction We define a Möbius ladder Mn of order n as the cubic circulant graph C2n(1; n) with 2n vertices. In this case, Mn can be considered as a regular 2n-gon whose n pairs of opposite vertices are joint by an edge. One can also realize Mn as a ladder with n steps on the Möbius band. A Prism graph Prn, sometimes also called a circular ladder graph, is a cubic graph with 2n vertices, which are connected as shown in Fig. 1. Notice, that Prism graph Pr2n is a double cover of Möbius ladder Mn. It is a discrete version of the statement that the cylinder is a double cover of the Möbius band. The aim of the present paper is to find the structure of the Jacobian group of the Möbius ladder Mn and Prism graph Prn. The notion of the Jacobian group of a graph (also known as Picard group, sandpile group, critical group) was independently given by many authors [1–3, 5]. This is 117.
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