We prove several complexity and decidability results for automatic monoids: (i) there exists an automatic monoid with a P-complete word problem, (ii) there exists an automatic monoid such that the firstorder theory of the corresponding Cayley-graph is not elementary decidable, and (iii) there exists an automatic monoid such that reachability in the corresponding Cayley-graph is undecidable. Mor...

Let S be a set of transpositions generating the symmetric group Sn, where n ≥ 3. It is shown that if the girth of the transposition graph of S is at least 5, then the automorphism group of the Cayley graph Cay(Sn, S) is the direct product Sn×Aut(T (S)), where T (S) is the transposition graph of S; the direct factors are the right regular representation of Sn and the image of the left regular ac...

Given a set A ⊆ Z/NZ we may form a Cayley sum graph GA on vertex set Z/NZ by joining i to j if and only if i + j ∈ A. We investigate the extent to which performing this construction with a random set A simulates the generation of a random graph, proving that the clique number of GA is a.s. O(logN). This shows that Cayley sum graphs can furnish good examples of Ramsey graphs. To prove this resul...

Many models of genome rearrangement involve operations (e.g. inversions and translocations) that are self-inverse, and hence generate a group acting on the space of genomes. This gives a correspondence between genome arrangements and the elements of a group, and consequently, between evolutionary paths and walks on the Cayley graph. Many common methods for phylogeny reconstruction rely on calcu...

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? ...