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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: On Vertex, Edge, and Vertex-Edge Random Graphs. | On Vertex Edge and Vertex-Edge Random Graphs Elizabeth Beer James Allen FilE Center for Computing Sciences 17100 Science Drive Bowie MD 20715-4300 USA libby.beer@gmail.com Department of Applied Mathematics and Statistics The Johns Hopkins University 3400 N. Charles Street Baltimore MD 21218-2682 USA jimfill@jhu.edu Svante Janson Edward R. Scheinerman Department of Mathematics Uppsala University P.O. Box 480 SE-751 06 Uppsala Sweden svante.janson@math.uu.se Department of Applied Mathematics and Statistics The Johns Hopkins University 3400 N. Charles Street Baltimore MD 21218-2682 USA ers@jhu.edu Submitted Oct 13 2010 Accepted May 3 2011 Published May 16 2011 Mathematics Subject Classification 05C80 Abstract We consider three classes of random graphs edge random graphs vertex random graphs and vertex-edge random graphs. Edge random graphs are Erdos-Renyi random graphs vertex random graphs are generalizations of geometric random graphs and vertexedge random graphs generalize both. The names of these three types of random graphs describe where the randomness in the models lies in the edges in the vertices or in both. We show that vertex-edge random graphs ostensibly the most general of the three models can be approximated arbitrarily closely by vertex random graphs but that the two categories are distinct. 1 Introduction The classic random graphs are those of Erdos and Renyi 8 9 . In their model each edge is chosen independently of every other. The randomness inhabits the edges vertices simply serve as placeholders to which random edges attach. Elizabeth Beer s research on this paper begun while she was a Ph.D. student at The Johns Hopkins University was supported by a National Defense Science and Engineering Graduate Fellowship. Research supported by The Johns Hopkins University s Acheson J. Duncan Fund for the Advancement of Research in Statistics. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P110 1 Since the introduction of Erdos-Renyi random graphs many other .
