23 research outputs found
Automorphisms of Cayley graphs on generalised dicyclic groups
A graph is called a GRR if its automorphism group acts regularly on its
vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that
there are only two infinite families of finite groups that do not admit GRRs :
abelian groups and generalised dicyclic groups. Indeed, any Cayley graph on
such a group admits specific additional graph automorphisms that depend only on
the group. Recently, Dobson and the last two authors showed that almost all
Cayley graphs on abelian groups admit no automorphisms other than these obvious
necessary ones. In this paper, we prove the analogous result for Cayley graphs
on the remaining family of exceptional groups: generalised dicyclic groups.Comment: 18 page
Asymptotic enumeration of graphical regular representations
We estimate the number of graphical regular representations (GRRs) of a given
group with large enough order. As a consequence, we show that almost all finite
Cayley graphs have full automorphism groups 'as small as possible'. This
confirms a conjecture of Babai-Godsil-Imrich-Lovasz on the proportion of GRRs,
as well as a conjecture of Xu on the proportion of normal Cayley graphs, among
Cayley graphs of a given finite group
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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
Further Applications of Schur Rings to Produce GRRs for Dihedral Groups
This note is a continuation of postgraduate thesis research carried out by
the first author under the supervision of the second author at the University
of Malta. In that research we took a look at several results relating Schur
rings to sufficient conditions for GRRs and then applied those results to
produce numerical methods for constructing trivalent GRRs for dihedral groups
very quickly.Comment: arXiv admin note: substantial text overlap with arXiv:2401.0230