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

Asymptotic Normality of Almost Local Functionals in Conditioned Galton-Watson Trees

An additive functional of a rooted tree is a functional that can be calculated recursively as the sum of the values of the functional over the branches, plus a certain toll function. Janson recently proved a central limit theorem for additive functionals of conditioned Galton-Watson trees under the assumption that the toll function is local, i.e. only depends on a fixed neighbourhood of the root. We extend his result to functionals that are almost local, thus covering a wider range of functionals. Our main result is illustrated by two explicit examples: the (logarithm of) the number of matchings, and a functional stemming from a tree reduction process that was studied by Hackl, Heuberger, Kropf, and Prodinger

Some Extremal And Structural Problems In Graph Theory

This work considers three main topics. In Chapter 2, we deal with König-Egerváry graphs. We will give two new characterizations of König-Egerváry graphs as well as prove a related lower bound for the independence number of a graph. In Chapter 3, we study joint degree vectors (JDV). A problem arising from statistics is to determine the maximum number of non-zero elements of a JDV. We provide reasonable lower and upper bounds for this maximum number. Lastly, in Chapter 4 we study a problem in chemical graph theory. In particular, we characterize extremal cases for the number of maximal matchings in two linear polymers of chemical interest: the polyspiro chains and benzenoid chains. We also enumerate maximal matchings in several classes of these linear polymers and use the obtained results to determine the asymptotic behavior of these matchings