# Browsing by Subject "Graph"

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Item Open Access A Study of Edge Toric Ideals using Associated Graphs(2012-04-26) Shen, YingyiThis thesis studies properties of edge toric ideals and resolutions by analyzing the associated graphs of algebraic structures. It mainly focused on proving that the repeated edges in a graph wouldn't change some properties of its underlying algebraic structure. An application of this result is that when we study multi-edge graphs, we can simplify in nite numbers of graphs to a simple one by deleting all the repeated edges.Item Open Access Complex-Balanced Steady States of Chemical Reaction Networks that Contain an Eulerian Cycle(2012-04-25) Thielman, Daniel WilsonThis work pertains to chemical reaction networks and their equilibria, called steady-states. Our main result states that for a cyclic chemical reaction net- work, there is a straightforward characterization for when a complex balancing steady state exists. We then extend our results to chemical reaction networks consisting of a closed path traversing each directed edge exactly once.Item Open Access The Graph Cases of the Riemannian Positive Mass and Penrose Inequalities in All Dimensions(2011) Lam, Mau-Kwong GeorgeWe consider complete asymptotically flat Riemannian manifolds that are the graphs of smooth functions over $\mathbb R^n$. By recognizing the scalar curvature of such manifolds as a divergence, we express the ADM mass as an integral of the product of the scalar curvature and a nonnegative potential function, thus proving the Riemannian positive mass theorem in this case. If the graph has convex horizons, we also prove the Riemannian Penrose inequality by giving a lower bound to the boundary integrals using the Aleksandrov-Fenchel inequality. We also prove the ZAS inequality for graphs in Minkowski space. Furthermore, we define a new quasi-local mass functional and show that it satisfies certain desirable properties.