60 citations · 145 across the 38 of their papers we have counts for
18 papers · 1 filter
Infinite Paley graphs
Gareth A. Jones
Infinite analogues of the Paley graphs are constructed, based on uncountably many infinite but locally finite fields. Weil's estimate for character sums shows that they are all iso…
Edge-transitive embeddings of complete graphs
Gareth A. Jones
Building on earlier work of Biggs, James, Wilson and the author, and using the Graver-Watkins description of the 14 classes of edge-transitive maps, we complete the classification…
Unstable maps
Gareth A. Jones
A map which is non-orientable or has non-empty boundary has a canonical double cover which is orientable and has empty boundary. The map is called stable if every automorphism of t…
Automorphism groups of edge-transitive maps
Gareth A. Jones
For each of the 14 classes of edge-transitive maps described by Graver and Watkins, necessary and sufficient conditions are given for a group to be the automorphism group of a map,…
Cayley properties of merged Johnson graphs
Gareth A. Jones, Robert Jajcay
Extending earlier results of Godsil and of Dobson and Malnic on Johnson graphs, we characterise those merged Johnson graphs which are Cayley graphs, that is, which are…
Conjugacy classes of reflections of maps
Gareth A. Jones
This paper considers how many conjugacy classes of reflections a map can have, under various transitivity conditions. It is shown that for vertex- and for face-transitive maps ther…
Chiral covers of hypermaps
Gareth A. Jones
Generalising a conjecture of Singerman, it is shown that there exist orientably regular chiral hypermaps of every non-spherical type. The proof uses the representation theory of au…
Combinatorial categories and permutation groups
Gareth A. Jones
The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group Γ, with quotient group isomorp…
Elementary abelian regular coverings of Platonic maps, Case I: ordinary representations
Gareth A. Jones
We classify the orientably regular maps which are elementary abelian regular branched coverings of Platonic maps M, in the case where the covering group and the rotation group G of…
Classification of nonorientable regular embeddings of Hamming graphs
Gareth A. Jones, Young Soo Kwon
By a regular embedding of a graph K in a surface we mean a 2-cell embedding of K in a compact connected surface such that the automorphism group acts regularly on flags. In this pa…
Hypermaps and multiply quasiplatonic Riemann surfaces
Gareth A. Jones
Generalising an example by Girondo and Wolfart, we use finite group theory to construct Riemann surfaces admitting two or more regular dessins (i.e. orientably regular hypermaps) w…
Constructions of Chiral Polytopes of Small Rank
Antonio Breda D'Azevedo, Gareth A. Jones, Egon Schulte
An abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. The present…
Classification and Galois conjugacy of Hamming maps
Gareth A. Jones
We show that for each d>0 the d-dimensional Hamming graph H(d,q) has an orientably regular surface embedding if and only if q is a prime power p^e. If q>2 there are up to isomorphi…
Characterisations and Galois conjugacy of generalised Paley maps
Gareth A. Jones
A generalised Paley map is a Cayley map for the additive group of a finite field F, with a subgroup S=-S of the multiplicative group as generating set, cyclically ordered by powers…
Regular embeddings of complete bipartite graphs: classification and enumeration
Gareth A. Jones
The regular embeddings of complete bipartite graphs in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are d…
Maps admitting trialities but not dualities
Gareth A. Jones, Andrew Poulton
We use group theory to construct infinite families of maps on surfaces which are invariant under Wilson's map operations of order 3 but not under the operations of order 2, such as…
Hypermap operations of finite order
Gareth A. Jones, Daniel Pinto
Duality and chirality are examples of operations of order 2 on hypermaps. James showed that the groups of all operations on hypermaps and on oriented hypermaps can be identified wi…
Chirality Groups of Maps and Hypermaps
Antonio Breda d'Azevedo, Gareth Jones, Roman Nedela +1
Although the phenomenon of chirality appears in many investigations of maps and hypermaps no detailed study of chirality seems to have been carried out. Chirality of maps and hyper…