Critical groups of graphs and generalizations

摘要项目负责人：Lorenzini， Dino项目编号：DMS - 090216机构：美国佐治亚大学研究基金会题目：图的关键群与推广关于矩阵的数学知识有一个庞大的体系，特别是有两组不变量被广泛用于研究任何整数矩阵：矩阵的特征值和矩阵的不变量因子。图的皮卡德群是一个可以用图的拉普拉斯算子的第二组不变量完全描述的对象。它的研究是在大约20年前由几位有着广泛不同观点的研究人员独立发起的，比如物理学家、算术几何学家和图论学家。本研究将探讨Laplacian的特征值如何影响Picard群的结构，以及最近关于图的Riemann-Roch定理是否可以推广到更大的整数格类。图形被用来模拟实际生活中自然发生的许多不同现象，例如电话网络或电路。与图自然相关的是一个称为图的拉普拉斯函数的数组，图可以从这个数组中完全恢复。寻找该数组的代数性质与图的组合性质之间的关系是本研究的主题之一。

ABSTRACTPrincipal Investigator: Lorenzini, Dino Proposal Number: DMS - 0902161Institution: University of Georgia Research Foundation IncTitle: Critical groups of graphs and generalizationsThere is a vast body of mathematical knowledge concerning matrices and, in particular, two sets of invariants are widely used to study any integer matrix: the eigenvalues of the matrix, and the invariant factors of the matrix. The Picard group of a graph is an object that can be completely described in terms of the second set of invariants of the Laplacian of the graph. Its study was independently initiated almost 20 years ago by several researchers with widely different points of views, such as physicists, arithmetic geometers, and graph theorists. This research will investigate how the eigenvalues of the Laplacian affect the structure of the Picard group, and whether the recent Riemann-Roch theorem for graphs can be extended to a larger class of integer lattices.Graphs are used to model many different phenomenons naturally occurring in practical life, such as telephone networks, or electrical circuits. Naturally associated to a graph is a array called the Laplacian of the graph, from which the graph can be completely recovered. Finding relationships between the algebraic properties of this array and the combinatorial properties of the graph is one of the main theme of this research.