In this paper we prove a Feynman-Kac-Itô formula for magnetic Schrödi-nger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation. On the operator side we identify a very general class of potentials that allows the definition of magnetic Schrödinger operators. On the probabilistic side, we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials. Apart from linking the world of discrete magnetic operators with the probabilistic world through the Feynman-Kac-Itô formula, the insights from this paper gained on both sides should be of an independent interest. As applications of the Feynman-Kac-Itô formula, we prove a Kato inequality, a Golden-Thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials. Mathematics Subject Classification

Abstract. We study a special class of graphs with a strong transience feature called uniform transience. We characterize uniform transience via a Feller-type property and via validity of an isoperimetric inequality. We then give a further characterization via equality of the Royden boundary and the harmonic boundary and show that the Dirichlet problem has a unique solution for such graphs. The Markov semigroups and resolvents (with Dirichlet boundary conditions) on these graphs are shown to be ultracontractive. Moreover, if the underlying measure is finite, the semigroups and resolvents are trace class and their generators have ℓ p independent pure point spectra (for 1 ≤ p ≤ ∞). Examples of uniformly transient graphs include Cayley graphs of hyperbolic groups as well as trees and Euclidean lattices of dimension at least three. As a surprising consequence, the Royden compactification of such lattices turns out to be the one-point compacitifcation and the Laplacians of such lattices have pure point spectrum if the underlying measure is chosen to be finite.

Abstract. We provide an introduction to Dirichlet forms on discrete spaces and study their global properties such as recurrence, stochastic completeness and regularity of the Neumann form. In this setting we compare the notion of a recurrent Dirichlet form and a recurrent discrete time Markov chain of a given graph. We prove several known and several new characterizations of recurrence by using functional analytic Dirichlet form methods only. Finally, we compare all the mentioned global properties and discuss their relation to spectral theory.

Abstract:We complete the classification of all symmetric designs of order nine admitting an automorphism of order six. As a matter of fact, the classification for the parameters (35,17,8), (56,11,2), and (91,10,1) had already been done, and in this paper we present the results for the parameters (36,15,6), (40,13,4), and (45,12,3). We also provide information about the order and the structure of the full automorphism groups of the constructed designs. # 2005 Wiley Periodicals, Inc.

We describe the set of all Dirichlet forms associated to a given infinite graph in terms of Dirichlet forms on its Royden boundary. Our approach is purely analytical and uses form methods.