Graphs with many valencies and few eigenvalues

Dom de Caen posed the question whether connected graphs with three distinct eigenvalues have at most three distinct valencies. We do not answer this question, but instead construct connected graphs with four and five distinct eigenvalues and arbitrarily many distinct valencies. The graphs with four distinct eigenvalues come from regular two-graphs. As a side result, we characterize the disconnected graphs and the graphs with three distinct eigenvalues in the switching class of a regular two-graph

Exact sampling and counting for fixed-margin matrices

The uniform distribution on matrices with specified row and column sums is often a natural choice of null model when testing for structure in two-way tables (binary or nonnegative integer). Due to the difficulty of sampling from this distribution, many approximate methods have been developed. We will show that by exploiting certain symmetries, exact sampling and counting is in fact possible in many nontrivial real-world cases. We illustrate with real datasets including ecological co-occurrence matrices and contingency tables.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1131 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1104.032

Universal Adjacency Matrices with Two Eigenvalues

AMS Mathematics Subject Classification: 05C50.Adjacency matrix;Universal adjacency matrix;Laplacian matrix;signless Laplacian;Graph spectra;Eigenvalues;Strongly regular graphs

Combinatorial Designs with Two Singular Values II. Partial Geometric Designs

In this and an earlier paper [17] we study combinatorial designs whose incidence matrix has two distinct singular values.These generalize (v, k, ë) designs, and include uniform multiplicative designs and partial geometric designs.Here we study the latter, which are precisely the designs with constant replication and block size.We collect most known results, give new characterization results, and we enumerate, partly by computer, all small ones.matrices;singularities;geometry;combinatorics