14,265 research outputs found
Antipodal Distance Transitive Covers of Complete Graphs
AbstractA distance-transitive antipodal cover of a complete graphKnpossesses an automorphism group that acts 2-transitively on the fibres. The classification of finite simple groups implies a classification of finite 2-transitive permutation groups, and this allows us to determine all possibilities for such a graph. Several new infinite families of distance-transitive graphs are constructed
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
ON A CLASS OF EDGE-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS
The paper is devoted to the problem of classification of edge-transitive distance-regular antipodal covers of complete graphs. This extends the classification of those covers that are arc-transitive, which has been settled except for some tricky cases that remain to be considered, including the case of covers satisfying condition (which means that every two vertices at distance 2 have exactly one common neighbour).Here it is shown that an edge-transitive distance-regular antipodal cover of a complete graph with is either the second neighbourhood of a vertex in a Moore graph of valency 3 or 7, or a Mathon graph, or a half-transitive graph whose automorphism group induces an affine -homogeneous group on the set of its fibres. Moreover, distance-regular antipodal covers of complete graphs with that admit an automorphism group acting -homogeneously on the set of fibres (which turns out to be an approximation of the property of edge-transitivity of such cover), are described. A well-known correspondence between distance-regular antipodal covers of complete graphs with and geodetic graphs of diameter two that can be viewed as underlying graphs of certain Moore geometries, allows us to effectively restrict admissible automorphism groups of covers under consideration by combining Kantor's classification of involutory automorphisms of these geometries together with the classification of finite 2-homogeneous permutation groups
Subdegree growth rates of infinite primitive permutation groups
A transitive group of permutations of a set is primitive if the
only -invariant equivalence relations on are the trivial and
universal relations.
If , then the orbits of the stabiliser on
are called the -suborbits of ; when acts transitively
the cardinalities of these -suborbits are the subdegrees of .
If acts primitively on an infinite set , and all the suborbits of
are finite, Adeleke and Neumann asked if, after enumerating the subdegrees
of as a non-decreasing sequence , the subdegree
growth rates of infinite primitive groups that act distance-transitively on
locally finite distance-transitive graphs are extremal, and conjecture there
might exist a number which perhaps depends upon , perhaps only on ,
such that .
In this paper it is shown that such an enumeration is not desirable, as there
exist infinite primitive permutation groups possessing no infinite subdegree,
in which two distinct subdegrees are each equal to the cardinality of
infinitely many suborbits. The examples used to show this provide several novel
methods for constructing infinite primitive graphs.
A revised enumeration method is then proposed, and it is shown that, under
this, Adeleke and Neumann's question may be answered, at least for groups
exhibiting suitable rates of growth.Comment: 41 page
On a Class of Edge-Transitive Distance-Regular Antipodal Covers of Complete Graphs
The paper is devoted to the problem of classification of edge-transitive distance-regular antipodal covers of complete graphs. This extends the classification of those covers that are arc-transitive, which has been settled except for some tricky cases that remain to be considered, including the case of covers satisfying condition c2=1 (which means that every two vertices at distance 2 have exactly one common neighbour). Here it is shown that an edge-transitive distance-regular antipodal cover of a complete graph with c2=1 is either the second neighbourhood of a vertex in a Moore graph of valency 3 or 7, or a Mathon graph, or a half-transitive graph whose automorphism group induces an affine 2-homogeneous group on the set of its fibres. Moreover, distance-regular antipodal covers of complete graphs with c2=1 that admit an automorphism group acting 2-homogeneously on the set of fibres (which turns out to be an approximation of the property of edge-transitivity of such cover), are described. A well-known correspondence between distance-regular antipodal covers of complete graphs with c2=1 and geodetic graphs of diameter two that can be viewed as underlying graphs of certain Moore geometries, allows us to effectively restrict admissible automorphism groups of covers under consideration by combining Kantor's classification of involutory automorphisms of these geometries together with the classification of finite 2-homogeneous permutation groups.This work was supported by the Russian Science Foundation under grant no. 20-71-00122
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