Morris, Joy

Permanent URI for this collection

https://opus.uleth.ca/handle/10133/4941

Browse

Recent Submissions

Now showing 1 - 5 of 29
  • Thumbnail Image
    Item
    On the automorphism groups of almost all circulant graphs and digraphs
    (University of Primorska, 2014) Bhoumik, Soumya; Dobson, Edward; Morris, Joy
    We attempt to determine the structure of the automorphism group of a generic circulant graph. We first show that almost all circulant graphs have automorphism groups as small as possible. The second author has conjectured that almost all of the remaining circulant (di)graphs (those whose automorphism group is not as small as possible) are normal circulant (di)graphs. We show this conjecture is not true in general, but is true if we consider only those circulant (di)graphs whose order is in a "large" subset of integers. We note that all non-normal circulant (di)graphs can be classified into two natural classes (generalized wreath products, and deleted wreath type), and show that neither of these classes contains almost every non-normal circulant digraph.
  • Thumbnail Image
    Item
    Most rigid representations and Cayley index
    (University of Primorska, 2018) Morris, Joy; Tymburski, Josh
    For any finite group G, a natural question to ask is the order of the smallest possible automorphism group for a Cayley graph on G. A particular Cayley graph whose automorphism group has this order is referred to as an MRR (Most Rigid Representation), and its Cayley index is a numerical indicator of this value. Study of GRRs showed that with the exception of two infinite families and thirteen individual groups, every group admits a Cayley graph whose MRR is a GRR, so that the Cayley index is 1. The full answer to the question of finding the smallest possible Cayley index for a Cayley graph on a fixed group was almost completed in previous work, but the precise answers for some finite groups and one infinite family of groups were left open. We fill in the remaining gaps to completely answer this question.
  • Thumbnail Image
    Item
    Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups
    (University of Primorska, 2020) Morgan, Luke; Morris, Joy; Verret, Gabriel
    Let Γ = Cay(G, S) be a Cayley digraph on a group G and let A = Aut(Γ). The Cayley index of Γ is |A : G|. It has previously been shown that, if p is a prime, G is a cyclic p-group and A contains a noncyclic regular subgroup, then the Cayley index of Γ is superexponential in p. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if p is an odd prime and G is abelian but not cyclic, and has order a power of p at least p3, then there is a Cayley digraph Γ on G whose Cayley index is just p, and whose automorphism group contains a nonabelian regular subgroup.
  • Thumbnail Image
    Item
    Two new families of non-CCA groups
    (University of Primorska, 2021) Fuller, Brandon; Morris, Joy
    We determine two new infinite families of Cayley graphs that admit colour-preserving automorphisms that do not come from the group action. By definition, this means that these Cayley graphs fail to have the CCA (Cayley Colour Automorphism) property, and the corresponding infinite families of groups also fail to have the CCA property. The families of groups consist of the direct product of any dihedral group of order 2n where n ≥ 3 is odd, with either itself, or the cyclic group of order n. In particular, this family of examples includes the smallest non-CCA group that does not fit into any previous family of known non-CCA groups.
  • Thumbnail Image
    Item
    On the asymptotic enumeration of Cayley graphs
    (Springer, 2021) Morris, Joy; Moscatiello, Mariapia; Spiga, Pablo
    In this paper, we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible auto- morphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is com- plicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing sepa- rately the cases where the regular group R has a nontrivial proper normal subgroup N with the property that the automorphism group of the digraph fixes each N-coset setwise, and the cases where it does not. In this paper, we deal with undirected graphs in the case where the regular group has such a nontrivial proper normal subgroup.