Projects / Programmes

# The Weiss Conjecture and Generalisations

Code |
Science |
Field |
Subfield |

1.01.05 |
Natural sciences and mathematics |
Mathematics |
Graph theory |

Code |
Science |
Field |

P001 |
Natural sciences and mathematics |
Mathematics |

Code |
Science |
Field |

1.01 |
Natural Sciences |
Mathematics |

mathematics, symmetry, graph, group, Weiss conjecture

##
Researchers (17)

##
Organisations (3)

##
Abstract

The topic of the proposed project belongs to the intersection of group theory and combinatorics (graph theory in particular). It deals with a fundamental mathematical questions: How symmetric can a certain mathematical object be? This is a very vague question and requires a more specific setting if specific answers are asked for. This question has attracted a lot of attention in the setting of discrete structures---graphs in particular, starting with the ingenious work of Tutte on cubic arc-transitive graphs and culminating in a deep and long-standing conjecture of Richard Weiss about the order of the automorphism group of a finite connected locally primitive arc-transitive graph.
The Weiss conjecture, stated in 1987, is one the most renown open problems in the algebraic graph theory and can be formulated as follows:
For every positive integer d there exists a constant c with the following property: If
X is a connected finite graph in which every vertex has valence d and G is a group of automorphisms of X that acts transitively on the ordered pairs of adjacent vertices of X and such that the permutation group induced by the action of the stabiliser H of a vertex v on its neighbourhood is primitive, then the order of H is at most c.
Several deep and long papers have been published since then, each proving a specific case of the conjecture, but it seems that the attempts to prove the conjecture need fresh ideas.
The aim of the proposed project is to explore several new possible approaches towards the proof the Weiss conjecture. The first approach is closest to the classical methods but plans to utilise recently proved and deep results in the area of local analysis of finite groups. The second approach is completely new and is based on the positive solution of the restricted Burnside problem, which allows one to reduce the Weiss conjecture to two separate problems about the exponent and rank of the vertex-stabiliser when the local action thereof is primiitve. The third approach that we want to explore is by generalising to a more general question about the growth of the vertex-stabiliser with respect to the order of the graph.
While we fully acknowledge that proving this 30 years old conjecture is an ambitious goal which might not be achieved during the duration of the proposed research project, we believe that the tools that have recently become available to us and an excellent research team will at least allow us to make a considerable progress towards the final resolution of the conjecture and at the same time increase our understanding of the symmetry in graphs.

##
Significance for science

The theme of the proposed project is one of the universal notions in science: symmetry. Even though we plan to investigate this notion in a very concrete setting, the theory of graphs, any result that we prove here will shed new light on that notion in its full generality. It might raise new questions about symmetry in other areas of mathematics and suggest new lines of research.
More concretely, the proposed project is about one of the oldest and most renown conjecture in the area of algebraic graph theory. Its final resolution would represent a major contribution to this mathematical area and would resonate highly in the community.
Furthermore, there are several graph theoretical problems where existence of an upper bound on the order of the automorphism group of a graph is useful. Any partial result towards the Weiss conjecture and the suggested generalisations will provide new opportunities to tackle these problems.
Finally, the suggested generalisations and new approaches described in detail in the proposal will yield new techniques for studying symmetry in graphs; and since the topic is closely linked to the group theory, also for the study of finite group theory.
Our initial results in this area have already attracted considerable interest in the mathematical community, resulting in many invitations to present the results at several renown scientific meetings. We believe that the results arising from the proposed project will receive similar attention.

##
Significance for the country

The theme of the proposed project is one of the universal notions in science: symmetry. Even though we plan to investigate this notion in a very concrete setting, the theory of graphs, any result that we prove here will shed new light on that notion in its full generality. It might raise new questions about symmetry in other areas of mathematics and suggest new lines of research.
More concretely, the proposed project is about one of the oldest and most renown conjecture in the area of algebraic graph theory. Its final resolution would represent a major contribution to this mathematical area and would resonate highly in the community.
Furthermore, there are several graph theoretical problems where existence of an upper bound on the order of the automorphism group of a graph is useful. Any partial result towards the Weiss conjecture and the suggested generalisations will provide new opportunities to tackle these problems.
Finally, the suggested generalisations and new approaches described in detail in the proposal will yield new techniques for studying symmetry in graphs; and since the topic is closely linked to the group theory, also for the study of finite group theory.
Our initial results in this area have already attracted considerable interest in the mathematical community, resulting in many invitations to present the results at several renown scientific meetings. We believe that the results arising from the proposed project will receive similar attention.