A Note on Perfect Slice Sampling

Perfect slice sampling is a method to turn Markov Chain Monte Carlo (MCMC) samplers into exact generators for independent random variates. We show that the simplest version of the perfect slice sampler suggested in the literature does not always sample from the target distribution. (author's abstract)Series: Research Report Series / Department of Statistics and Mathematic

A Note on the Folding Coupler

Perfect Gibbs sampling is a method to turn Markov Chain Monte Carlo (MCMC) samplers into exact generators for independent random vectors. We show that a perfect Gibbs sampling algorithm suggested in the literature is not always generating from the correct distribution. (author's abstract)Series: Research Report Series / Department of Statistics and Mathematic

Algebraic Connectivity and Degree Sequences of Trees

We investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence. We show that such trees are caterpillars and that the vertex degrees are non-decreasing on every path on non-pendant vertices starting at the characteristic set of the Fiedler vector.Comment: 8 page

A Faber-Krahn-type Inequality for Regular Trees

In the last years some results for the Laplacian on manifolds have been shown to hold also for the graph Laplacian, e.g. Courant's nodal domain theorem or Cheeger's inequality. Friedman (Some geometric aspects of graphs and their eigenfunctions, Duke Math. J. 69 (3), pp. 487-525, 1993) described the idea of a ``graph with boundary". With this concept it is possible to formulate Dirichlet and Neumann eigenvalue problems. Friedman also conjectured another ``classical" result for manifolds, the Faber-Krahn theorem, for regular bounded trees with boundary. The Faber-Krahn theorem states that among all bounded domains $D \subset R^n$ with fixed volume, a ball has lowest first Dirichlet eigenvalue. In this paper we show such a result for regular trees by using a rearrangement technique. We give restrictive conditions for trees with boundary where the first Dirichlet eigenvalue is minimized for a given "volume". Amazingly Friedman's conjecture is false, i.e. in general these trees are not ``balls". But we will show that these are similar to ``balls". (author's abstract)Series: Preprint Series / Department of Applied Statistics and Data Processin

rstream: Streams of Random Numbers for Stochastic Simulation

The package rstream provides a unified interface to streams of random numbers for the R statistical computing language. Features are: * independent streams of random numbers * substreams * easy handling of streams (initialize, reset) * antithetic random variates The paper describes this packages and demonstrates an simple example the usefulness of this approach.Series: Preprint Series / Department of Applied Statistics and Data Processin

Largest Laplacian Eigenvalue and Degree Sequences of Trees

We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization.Comment: 9 pages, 2 figure

Convex Cycle Bases

Convex cycles play a role e.g. in the context of product graphs. We introduce convex cycle bases and describe a polynomial-time algorithm that recognizes whether a given graph has a convex cycle basis and provides an explicit construction in the positive case. Relations between convex cycles bases and other types of cycles bases are discussed. In particular we show that if G has a unique minimal cycle bases, this basis is convex. Furthermore, we characterize a class of graphs with convex cycles bases that includes partial cubes and hence median graphs. (authors' abstract)Series: Research Report Series / Department of Statistics and Mathematic

Graph Laplacians, Nodal Domains, and Hyperplane Arrangements

Eigenvectors of the Laplacian of a graph G have received increasing attention in the recent past. Here we investigate their so-called nodal domains, i.e. the connected components of the maximal induced subgraphs of G on which an eigenvector ψ does not change sign. An analogue of Courant's nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of ψ in the spectrum. This bound, however, is not sharp in general. In this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph. The class of Boolean Hypercubes is discussed in detail. We find that, despite the simplicity of this graph class, for which complete spectral information is available, the computations are still non-trivial. Nevertheless, we obtained some new results and a number of conjectures

A note on quasi-robust cycle bases

We investigate here some aspects of cycle bases of undirected graphs that allow the iterative construction of all elementary cycles. We introduce the concept of quasi-robust bases as a generalization of the notion of robust bases and demonstrate that a certain class of bases of the complete bipartite graphs K m,n with m,n _> 5 is quasi-robust but not robust. We furthermore disprove a conjecture for cycle bases of Cartesian product graphs