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

Sets as graphs

The aim of this thesis is a mutual transfer of computational and structural results and techniques between sets and graphs. We study combinatorial enumeration of sets, canonical encodings, random generation, digraph immersions. We also investigate the underlying structure of sets in algorithmic terms, or in connection with hereditary graphs classes. Finally, we employ a set-based proof-checker to verify two classical results on claw-free graph

Distance Degree Regular Graphs and Theireccentric Digraphs

The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G.The distance degree sequence (dds) of a vertex v in a graph G = (V,E) is a list of the number of vertices at distance 1, 2, . . . , e(u) in that order, where e(u) denotes the eccentricity of v in G. Thus the sequence (di0 , di1 , di2 , . . . , dij , . . .) is the dds of the vertex vi in G where dij denotes number of vertices at distance j from vi. A graph is distance degree regular (DDR) graph if all vertices have the same dds. A vertex v is an eccentric vertex of vertex u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a graph (digraph) G is the digraph that has the same vertex as G and an arc from u to v exists in ED(G) if and only if v is an eccentric vertex of u in G. In this paper, we consider the construction of new families of DDR graphs with arbitrary diameter. Also we consider some special class of DDR graphs in relation with eccentric digraph of a graph. Different structural properties of eccentric digraphs of DDR graphs are dealt herewith

Distance Degree Regular Graphs and Distance Degree Injective Graphs: An Overview

The distance d ( v , u ) from a vertex v of G to a vertex u is the length of shortest v to u path. The eccentricity e v of v is the distance to a farthest vertex from v . If d ( v , u ) = e ( v ) , ( u ≠ v ) , we say that u is an eccentric vertex of v . The radius rad ( G ) is the minimum eccentricity of the vertices, whereas the diameter diam ( G ) is the maximum eccentricity. A vertex v is a central vertex if e ( v ) = r a d ( G ) , and a vertex is a peripheral vertex if e ( v ) = d i a m ( G ) . A graph is self-centered if every vertex has the same eccentricity; that is, r a d ( G ) = d i a m ( G ) . The distance degree sequence (dds) of a vertex v in a graph G = ( V , E ) is a list of the number of vertices at distance 1 , 2 , . . . . , e ( v ) in that order, where e ( v ) denotes the eccentricity of v in G . Thus, the sequence ( d i 0 , d i 1 , d i 2 , … , d i j , … ) is the distance degree sequence of the vertex v i in G where d i j denotes the number of vertices at distance j from v i . The concept of distance degree regular (DDR) graphs was introduced by Bloom et al., as the graphs for which all vertices have the same distance degree sequence. By definition, a DDR graph must be a regular graph, but a regular graph may not be DDR. A graph is distance degree injective (DDI) graph if no two vertices have the same distance degree sequence. DDI graphs are highly irregular, in comparison with the DDR graphs. In this paper we present an exhaustive review of the two concepts of DDR and DDI graphs. The paper starts with an insight into all distance related sequences and their applications. All the related open problems are listed

Diameter, Girth And Other Properties Of Highly Symmetric Graphs

We consider a number of problems in graph theory, with the unifying theme being the properties of graphs which have a high degree of symmetry. In the degree-diameter problem, we consider the question of finding asymptotically large graphs of given degree and diameter. We improve a number of the current best published results in the case of Cayley graphs of cyclic, dihedral and general groups. In the degree-diameter problem for mixed graphs, we give a new corrected formula for the Moore bound and show non-existence of mixed Cayley graphs of diameter 2 attaining the Moore bound for a range of open cases. In the degree-girth problem, we investigate the graphs of Lazebnik, Ustimenko and Woldar which are the best asymptotic family identified to date. We give new information on the automorphism groups of these graphs, and show that they are more highly symmetrical than has been known to date. We study a related problem in group theory concerning product-free sets in groups, and in particular those groups whose maximal product-free subsets are complete. We take a large step towards a classification of such groups, and find an application to the degree-diameter problem which allows us to improve an asymptotic bound for diameter 2 Cayley graphs of elementary abelian groups. Finally, we study the problem of graphs embedded on surfaces where the induced map is regular and has an automorphism group in a particular family. We give a complete enumeration of all such maps and study their properties