5,445 research outputs found
Countable locally 2-arc-transitive bipartite graphs
We present an order-theoretic approach to the study of countably infinite
locally 2-arc-transitive bipartite graphs. Our approach is motivated by
techniques developed by Warren and others during the study of cycle-free
partial orders. We give several new families of previously unknown countably
infinite locally-2-arc-transitive graphs, each family containing continuum many
members. These examples are obtained by gluing together copies of incidence
graphs of semilinear spaces, satisfying a certain symmetry property, in a
tree-like way. In one case we show how the classification problem for that
family relates to the problem of determining a certain family of highly
arc-transitive digraphs. Numerous illustrative examples are given.Comment: 29 page
Transitivity conditions in infinite graphs
We study transitivity properties of graphs with more than one end. We
completely classify the distance-transitive such graphs and, for all , the -CS-transitive such graphs.Comment: 20 page
On P-transitive graphs and applications
We introduce a new class of graphs which we call P-transitive graphs, lying
between transitive and 3-transitive graphs. First we show that the analogue of
de Jongh-Sambin Theorem is false for wellfounded P-transitive graphs; then we
show that the mu-calculus fixpoint hierarchy is infinite for P-transitive
graphs. Both results contrast with the case of transitive graphs. We give also
an undecidability result for an enriched mu-calculus on P-transitive graphs.
Finally, we consider a polynomial time reduction from the model checking
problem on arbitrary graphs to the model checking problem on P-transitive
graphs. All these results carry over to 3-transitive graphs.Comment: In Proceedings GandALF 2011, arXiv:1106.081
Extendable self-avoiding walks
The connective constant mu of a graph is the exponential growth rate of the
number of n-step self-avoiding walks starting at a given vertex. A
self-avoiding walk is said to be forward (respectively, backward) extendable if
it may be extended forwards (respectively, backwards) to a singly infinite
self-avoiding walk. It is called doubly extendable if it may be extended in
both directions simultaneously to a doubly infinite self-avoiding walk. We
prove that the connective constants for forward, backward, and doubly
extendable self-avoiding walks, denoted respectively by mu^F, mu^B, mu^FB,
exist and satisfy mu = mu^F = mu^B = mu^FB for every infinite, locally finite,
strongly connected, quasi-transitive directed graph. The proofs rely on a 1967
result of Furstenberg on dimension, and involve two different arguments
depending on whether or not the graph is unimodular.Comment: Accepted versio
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