Linearly bounded infinite graphs

Graphs with Linearly Bounded Ramsey Numbers

A note on bounded automorphisms of infinite graphs

Let X be a connected locally finite graph with vertex-transitive automorphism group. If X has polynomial growth then the set of all bounded automorphisms of finite order is a locally finite, periodic normal subgroup of AUT(X) and the action of AUT(X) on V(X) is imprimitive if X is not finite. If X has infinitely many ends, the group of bounded automorphisms itself is locally finite and periodic.

Geodesics and Bounded Harmonic Functions on Infinite Planar Graphs

It is shown there that an infinite connected planar graph with a uniform upper bound on vertex degree and rapidly decreasing Green's function (relative to the simple random walk) has infinitely many pairwise finitelyintersecting geodesic rays starting at each vertex. We then demonstrate the existence of nonconstant bounded harmonic functions on the graph. Let g be an infinite, simple, connected...

Spectral radius of finite and infinite planar graphs and of graphs of bounded genus

It is well known that the spectral radius of a tree whose maximum degree is D cannot exceed 2 √ D − 1. In this paper we derive similar bounds for arbitrary planar graphs and for graphs of bounded genus. It is proved that a the spectral radius ρ(G) of a planar graph G of maximum vertex degree D ≥ 4 satisfies √D ≤ ρ(G) ≤ √8D − 16 + 7.75. This result is best possible up to the additive constant—we...

Infinite Bounded Permutation Groups

This work concerns permutation groups on uncountable sets. The environment is ZFC, Set Theory with Axiom of Choice. Therefore the cardinal numbers are wellordered, a cardinal number k is infinite if and only if Ko < k, and each cardinal number k has a unique successor k. Throughout the paper Cl will be a set (usually infinite), n its cardinality, and m a cardinal number less than n. The support...