Morris, Joy

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

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Browsing Morris, Joy by Subject "Automorphism group"

Now showing 1 - 6 of 6
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    Groups for which it is easy to detect graphical regular representations
    (University of Primorska, 2022) Morris, Dave W.; Morris, Joy; Verret, Gabriel
    We say that a finite group G is DRR-detecting if, for every subset S of G, either the Cayley digraph Cay(G,S) is a digraphical regular representation (that is, its automorphism group acts regularly on its vertex set) or there is a nontrivial group automorphism φ of G such that φ(S) = S. We show that every nilpotent DRR-detecting group is a p-group, but that the wreath product Zp wr Zp is not DRR-detecting, for every odd prime p. We also show that if G1 and G2 are nontrivial groups that admit a digraphical regular representation and either gcd(|G1|, |G2|) = 1, or G2 is not DRR-detecting, then the direct product G1 x G2 is not DRR-detecting. Some of these results also have analogues for graphical regular representations.
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    Haar graphical representations of finite groups and an application to poset representations
    (Elsevier, 2025) Morris, Joy; Spiga, Pablo
    Answering a question of Feng, Kovács, Wang, and Yang, we classify the finite groups admitting a Haar graphical representation. Specifically, we show that every finite group admits a Haar graphical representation, with abelian groups and ten other small groups as the only exceptions. Our work on Haar graphs allows us to improve a 1980 result of Babai concerning representations of groups on posets, achieving the best possible result in this direction. An improvement to Babai's related result on representations of groups on distributive lattices follows.
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    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.
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    On the automorphism groups of almost all circulant graphs and digraphs
    (Drustvo Matematikov, Fizikov in Astronomov, 2018) 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.
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    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.
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    Strongly regular edge-transitive graphs
    (Drustvo Matematikov, Fizikov in Astronomov, 2009) Morris, Joy; Praeger, Cheryl E.; Spiga, Pablo
    In this paper, we examine the structure of vertex- and edge-transitive strongly regular graphs,using normal quotient reduction. We show that their reducible graphs in this family have quasi primitive automorphism groups, and prove (using the Classification of Finite Simple Groups) that no graph in this family has a holomorphic simple automorphism group. We also find some constraints on the parameters of the graphs in this family that reduce to complete graphs