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

Digraphs and homomorphisms: Cores, colorings, and constructions

A natural digraph analogue of the graph-theoretic concept of an `independent set\u27 is that of an acyclic set, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets

Infinite primitive and distance transitive directed graphs of finite out-valency

We give certain properties which are satisfied by the descendant set of a vertex in an infinite, primitive, distance transitive digraph of finite out-valency and provide a strong structure theory for digraphs satisfying these properties. In particular, we show that there are only countably many possibilities for the isomorphism type of such a descendant set, thereby confirming a conjecture of the first Author. As a partial converse, we show that certain related conditions on a countable digraph are sufficient for it to occur as the descendant set of a primitive, distance transitive digraph

Quantum Hall Ground States, Binary Invariants, and Regular Graphs

Extracting meaningful physical information out of a many-body wavefunction is often impractical. The polynomial nature of fractional quantum Hall (FQH) wavefunctions, however, provides a rare opportunity for a study by virtue of ground states alone. In this article, we investigate the general properties of FQH ground state polynomials. It turns out that the data carried by an FQH ground state can be essentially that of a (small) directed graph/matrix. We establish a correspondence between FQH ground states, binary invariants and regular graphs and briefly introduce all the necessary concepts. Utilizing methods from invariant theory and graph theory, we will then take a fresh look on physical properties of interest, e.g. squeezing properties, clustering properties, etc. Our methodology allows us to `unify' almost all of the previously constructed FQH ground states in the literature as special cases of a graph-based class of model FQH ground states, which we call \emph{accordion} model FQH states

Highly arc transitive digraphs

Unendliche, hochgradig bogentransitive Digraphen werden definiert und anhand von Beispielen vorgestellt. Die Erreichbarkeitsrelation und Eigenschaft–Z werden definiert und unter Verwendung von Knotengraden, Wachstum und anderen Eigenschaften, die von der Untersuchung von Nachkommen von Doppelstrahlen oder Automorphismengruppen herrühren, auf hochgradig bogentransitiven Digraphen untersucht. Seifters Theoreme über hochgradig bogentransitive Digraphen mit mehr als einem Ende, seine daherrührende Vermutung und deren sie widerlegende Gegenbeispiele werden vorgestellt. Eine Bedingung, unter der C–homogene Digraphen hochgradig bogentransitiv sind, wird angegeben und die Verbindung zwischen hochgradig bogentransitiven Digraphen und total unzusammenhängenden, topologischen Gruppen wird erwähnt. Einige Bemerkungen über die Vermutung von Cameron–Praeger–Wormald werden gemacht und eine verfeinerte Version vermutet. Die Eigenschaften der bekannten hochgradig bogentransitiven Digraphen werden gesammelt. Es wird festgestellt, dass einige, aber nicht alle unter ihnen Cayley–Graphen sind. Schließlich werden offen gebliebene Fragestellungen und Vermutungen zusammengefasst und neue hinzugefügt. Für die vorgestellten Lemmata, Propositionen und Theoreme sind entweder Beweise enthalten, oder Referenzen zu Beweisen werden angegeben.Infinite, highly arc transitive digraphs are defined and examples are given. The Reachability–Relation and Property-Z are defined and investigated on infinite, highly arc transitive digraphs using the valencies, spread and other properties arising from the investigation of the descendants of lines or the automorphism groups. Seifters theorems about highly arc transitive digraphs with more than one end, his conjecture on them and the counterexamples that disproved his conjecture, are given. A condition for C–homogeneous digraphs to be highly arc transitve is stated and the connection between highly arc transitive digraphs and totally disconnected, topological groups is mentioned. Some notes on the Cameron–Praeger–Wormald–Conjecture are made and a refined conjecture is stated. The properties of the known highly arc transitive digraphs are collected, some but not all of them are Cayley–graphs. Finally open questions and conjectures are stated and new ones are added. For the given lemmas, propositions and theorems either proofs or references to proofs are included

A graph theoretic proof of the complexity of colouring by a local tournament with at least two directed cycles

In this paper we give a graph theoretic proof of the fact that deciding whether a homomorphism exists to a fixed local tournament with at least two directed cycles is NP-complete. One of the main reasons for the graph theoretic proof is that it showcases all of the techniques that have been built up over the years in the study of the digraph homomorphism problem