Random Threshold Graphs

We introduce a pair of natural, equivalent models for random threshold graphs and use these models to deduce a variety of properties of random thre shold graphs. Specifically, a random threshold graph G is generated by choosing n IID values x1, . . . , xn uniformly in [0, 1]; distinct verticesi, j of G are adjacent exactly when xi + x j > 1. We examine various properties of random threshold graphs such as chromatic num ber, algebraic connectivity, and the existence of Hamiltonian cycles and perfect matchin gs.

[1]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[2]  R. Bhatia.  Positive Definite Matrices , 2007 .

[3]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[4]  J. Dall,et al.  Random geometric graphs. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  N. Mahadev,et al.  Threshold graphs and related topics , 1995 .

[6]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[7]  A. Hagberg,et al.  Designing threshold networks with given structural and dynamical properties. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  P. Hammer,et al.  Aggregation of inequalities in integer programming. , 1975 .

[9]  Edward R. Scheinerman,et al.  Random interval graphs , 1988, Comb..

[10]  C. Geiss,et al.  An introduction to probability theory , 2008 .

[11]  W. Hoeffding.  A Class of Statistics with Asymptotically Normal Distribution , 1948 .

[12]  Yechezkel Zalcstein,et al.  A Graph-Theoretic Characterization of the PV_chunk Class of Synchronizing Primitives , 1977, SIAM J. Comput..

[13]  F. Harary,et al.  Hamiltonian threshold graphs , 1987, Discret. Appl. Math..

[14]  Ing Rj Ser.  Approximation Theorems of Mathematical Statistics , 1980 .

[15]  Abdus Salam,et al.  Random Walks and Catalan Factorization , 2007 .

[16]  Robert Brawer,et al.  The linear algebra of the pascal matrix , 1992 .

[17]  Russell Merris,et al.  The Laplacian Spectrum of a Graph II , 1994, SIAM J. Discret. Math..