We show that on a Cayley graph of a nonamenable group, a.s. the infinite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has positive speed, and that these clusters admit bounded harmonic functions. A principal new finding on which these results are based is that such clusters admit invariant random subgraphs with positive isoper...

During a discussion taking place at WMC’01, G. Paun put the question of what could be computed only by moving symbols between membranes. In this paper we provide some elements of the answer, in a setting similar to tissue P systems, where the set of membranes is organized into a finite graph or into a Cayley graph, and using a very simple propagation process characterizing accretive growth. Our...

Let G be a group. We will call G a group with Cayley data if we are given all three of the following: the underlying set G; the collection of all Cayley sets of G; and for each Cayley subset S of G, the corresponding Cayley graph Cay(G,S). Is it then possible, from the Cayley data, to reconstruct the binary operation of the group? Is it possible to determine the isomorphism class of the group? ...

The connective constant μ(G) of an infinite transitive graph G is the exponential growth rate of the number of self-avoiding walks from a given origin. The relationship between connective constants and amenability is explored in the current work. Various properties of connective constants depend on the existence of so-called ‘unimodular graph height functions’, namely: (i) whether μ(G) is a loc...

A Cayley graph Cay(G,S) on a group G is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay(G,S). In this paper, all connected tetravalent non-normal Cayley graphs of order 4p are constructed explicitly for each prime p. As a result, there are fifteen sporadic and eleven infinite families of tetravalent non-normal Cayley graphs of orde...

In this paper, we study percolation on finite Cayley graphs. A conjecture of Benjamini says that the critical percolation pc of any vertex–transitive graph satisfying a certain diameter condition can be bounded away from one. We prove Benjamini’s conjecture for some special classes of Cayley graphs. We also establish a reduction theorem, which allows us to build Cayley graphs for large groups w...

This thesis investigates uniform mixing on Cayley graphs over Z3. We apply Mullin’s results on Hamming quotients, and characterize the 2(d+2)-regular connected Cayley graphs over Z3 that admit uniform mixing at time 2π/9. We generalize Chan’s construction on the Hamming scheme H(d, 2) to the schemeH(d, 3), and find some distance graphs of the Hamming graph H(d, 3) that admit uniform mixing at t...

Let Cay(H, S), S/H be an arbitrary Cayley graph over a finite group H. We shall say that two Cayley graphs Cay(H, S) and Cay(H, T) are Cayley isomorphic if there exists an automorphism . # Aut(H ) such that S=T. It is a trivial observation that two Cayley isomorphic Cayley graphs are isomorphic as graphs. The converse is not true: two Cayley graphs over the same group may be isomorphic as graph...