Critical groups of graphs and generalizations

主要研究人员：Lorenzini，Dino建议编号：DMS-0902161机构：佐治亚大学研究基金会Inc标题：临界图群和推广有大量关于矩阵的数学知识，特别是，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Picard群是可以用图的拉普拉斯不变量的第二不变量集完全描述的对象。这项研究是近20年前由几位观点迥异的研究人员独立发起的，这些研究人员包括物理学家、算术几何学家和图论学家。本研究将研究拉普拉斯本征值如何影响Picard群的结构，以及最近关于图的Riemann-Roch定理是否可以扩展到更大类的整数格。图被用来模拟实际生活中自然发生的许多不同的现象，如电话网络或电路。与图自然关联的是称为图的拉普拉斯的数组，从该数组可以完全恢复图。寻找这种数组的代数性质和图的组合性质之间的关系是本研究的主要主题之一。

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.