831 research outputs found
Locally -distance transitive graphs
We give a unified approach to analysing, for each positive integer , a
class of finite connected graphs that contains all the distance transitive
graphs as well as the locally -arc transitive graphs of diameter at least
. A graph is in the class if it is connected and if, for each vertex ,
the subgroup of automorphisms fixing acts transitively on the set of
vertices at distance from , for each from 1 to . We prove that
this class is closed under forming normal quotients. Several graphs in the
class are designated as degenerate, and a nondegenerate graph in the class is
called basic if all its nontrivial normal quotients are degenerate. We prove
that, for , a nondegenerate, nonbasic graph in the class is either a
complete multipartite graph, or a normal cover of a basic graph. We prove
further that, apart from the complete bipartite graphs, each basic graph admits
a faithful quasiprimitive action on each of its (1 or 2) vertex orbits, or a
biquasiprimitive action. These results invite detailed additional analysis of
the basic graphs using the theory of quasiprimitive permutation groups.Comment: Revised after referee report
Disimplicial arcs, transitive vertices, and disimplicial eliminations
In this article we deal with the problems of finding the disimplicial arcs of
a digraph and recognizing some interesting graph classes defined by their
existence. A diclique of a digraph is a pair of sets of vertices such
that is an arc for every and . An arc is
disimplicial when is a diclique. We show that the problem
of finding the disimplicial arcs is equivalent, in terms of time and space
complexity, to that of locating the transitive vertices. As a result, an
efficient algorithm to find the bisimplicial edges of bipartite graphs is
obtained. Then, we develop simple algorithms to build disimplicial elimination
schemes, which can be used to generate bisimplicial elimination schemes for
bipartite graphs. Finally, we study two classes related to perfect disimplicial
elimination digraphs, namely weakly diclique irreducible digraphs and diclique
irreducible digraphs. The former class is associated to finite posets, while
the latter corresponds to dedekind complete finite posets.Comment: 17 pags., 3 fig
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
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