Critical groups of graphs and generalizations

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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.