Báo cáo toán học: " Random Threshold Graphs Elizabeth Perez Reilly Edward R. Scheinerman"

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: Random Threshold Graphs Elizabeth Perez Reilly Edward R. Scheinerman. | Random Threshold Graphs Elizabeth Perez Reilly Edward R. Scheinerman Department of Applied Mathematics and Statistics Johns Hopkins University Baltimore Maryland 21218 USA. Submitted Feb 3 2009 Accepted Oct 13 2009 Published Oct 31 2009 Mathematics Subject Classifications 05C62 05C80 Abstract We introduce a pair of natural equivalent models for random threshold graphs and use these models to deduce a variety of properties of random threshold graphs. Specifically a random threshold graph G is generated by choosing n IID values x1 . xn uniformly in 0 1 distinct vertices i j of G are adjacent exactly when Xị Xj 1. We examine various properties of random threshold graphs such as chromatic number algebraic connectivity and the existence of Hamiltonian cycles and perfect matchings. 1 Introduction and Overview of Results Threshold graphs were introduced by Chvatal and Hammer in 4 5 see also 6 13 . There are several logically equivalent ways to define this family of graphs but the one we choose works well for developing a model of random graphs. A simple graph G is a threshold graph if we can assign weights to the vertices such that a pair of distinct vertices is adjacent exactly when the sum of their assigned weights is or exceeds a specified threshold. Without loss of generality the threshold can be taken to be 1 and the weights can be restricted to lie in the interval 0 1 see Definition . References 2 9 16 provide an extensive introduction to this class of graphs. If we choose the weights for the vertices at random we induce a probability measure on the set of threshold graphs and thereby create a notion of a random threshold graph. Given that we may assume the weights lie in 0 1 it is natural to take the weights independently and uniformly in that interval a careful definition is given in . The idea of choosing a random representation has been explored in other contexts such as random geometric graphs 18 choose points in a metric space at random to represent .