143 research outputs found
Classification of some countable descendant-homogeneous digraphs
For finite q, we classify the countable, descendant-homogeneous digraphs in
which the descendant set of any vertex is a q-valent tree. We also give
conditions on a rooted digraph G which allow us to construct a countable
descendant-homogeneous digraph in which the descendant set of any vertex is
isomorphic to G.Comment: 16 page
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
Constructing continuum many countable, primitive, unbalanced digraphs
AbstractWe construct continuum many non-isomorphic countable digraphs which are highly arc transitive, have finite out-valency and infinite in-valency, and whose automorphism groups are primitive
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
Species, Clusters and the 'Tree of Life': A graph-theoretic perspective
A hierarchical structure describing the inter-relationships of species has
long been a fundamental concept in systematic biology, from Linnean
classification through to the more recent quest for a 'Tree of Life.' In this
paper we use an approach based on discrete mathematics to address a basic
question: Could one delineate this hierarchical structure in nature purely by
reference to the 'genealogy' of present-day individuals, which describes how
they are related with one another by ancestry through a continuous line of
descent? We describe several mathematically precise ways by which one can
naturally define collections of subsets of present day individuals so that
these subsets are nested (and so form a tree) based purely on the directed
graph that describes the ancestry of these individuals. We also explore the
relationship between these and related clustering constructions.Comment: 19 pages, 4 figure
- …