Subcritical graph classes containing all planar graphs

We construct minor-closed addable families of graphs that are subcritical and contain all planar graphs. This contradicts (one direction of) a well-known conjecture of Noy

Compositions into Powers of bb: Asymptotic Enumeration and Parameters

For a fixed integer base $b\geq2$, we consider the number of compositions of $1$ into a given number of powers of $b$ and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of $b$. We study the asymptotic growth of those numbers and give precise asymptotic formulae for them, thereby improving on earlier results of Molteni. Our approach uses generating functions, which we obtain from infinite transfer matrices. With the same techniques the distribution of the largest denominator and the number of distinct parts are investigated

Free Lamplighter Groups and a Question of Atiyah

We compute the von Neumann dimensions of the kernels of adjacency operators on free lamplighter groups and show that they are irrational, thus providing an elementary constructive answer to a question of Atiyah.Comment: AMSLaTeX, 10 pages, to appear in Amer J Mat

The shape of random tanglegrams

A tanglegram consists of two binary rooted trees with the same number of leaves and a perfect matching between the leaves of the trees. We show that the two halves of a random tanglegram essentially look like two independently chosen random plane binary trees. This fact is used to derive a number of results on the shape of random tanglegrams, including theorems on the number of cherries and generally occurrences of subtrees, the root branches, the number of automorphisms, and the height. For each of these, we obtain limiting probabilities or distributions. Finally, we investigate the number of matched cherries, for which the limiting distribution is identified as well