# Faculty Research and Publications

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### Browsing Faculty Research and Publications by Subject "Automorphism"

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- ItemAutomorphism groups of wreath product digraphs(Electronic Journal of Combinatorics, 2009) Dobson, Edward; Morris, JoyWe generalize a classical result of Sabidussi that was improved by Hemminger, to the case of directed color graphs. The original results give a necessary and suﬃcient condition on two graphs, C and D, for the automorphsim group of the wreath product of the graphs, Aut(C o D) to be the wreath product of the automorphism groups Aut(C) o Aut(D). Their characterization generalizes directly to the case of color graphs, but we show that there are additional exceptional cases in which either C or D is an inﬁnite directed graph. Also, we determine what Aut(C o D) is if Aut(C o D) 6= Aut(C) o Aut(D), and in particular, show that in this case there exist vertex-transitive graphs C0 and D0 such that C0 oD0 = C oD and Aut(C oD) = Aut(C0) o Aut(D0).
- ItemAutomorphisms of circulants that respect partitions(University of Calgary, Department of Mathematics & Statistics, 2016) Morris, JoyIn this paper, we begin by partitioning the edge (or arc) set of a circulant (di)graph according to which generator in the connection set leads to each edge. We then further reﬁne the partition by subdividing any part that corresponds to an element of order less than n, according to which of the cycles generated by that element the edge is in. It is known that if the (di)graph is connected and has no multiple edges, then any automorphism that respects the ﬁrst partition and ﬁxes the vertex corresponding to the group identity must be an automorphism of the group (this is in fact true in the more general context of Cayley graphs). We show that automorphisms that respect the second partition and ﬁx 0 must also respect the ﬁrst partition, and so are again precisely the group automorphisms of Zn.
- ItemOn colour-preserving automorphisms of Caley graphs(Drustvo Matematikov, Fizikov in Astronomov, 2016) Hujdurovic, Ademir; Kutnar, Klavdija; Morris, Dave Witte; Morris, JoyWe study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups G, such that every such automorphism of every connected Cayley graph on G has a very simple form: the composition of a left-translation and a group automorphism. We ﬁnd classes of groups that have the property, and we determine the orders of all groups that do not have the property. We also have analogous results for automorphisms that permute the colours, rather than preserving them.
- ItemToida's conjecture is true(Electronic Journal of Combinatorics, 2002) Dobson, Edward; Morris, JoyLet S be a subset of the units in Zn. Let Γ be a circulant graph of order n (a Cayley graph of Zn) such that if ij ∈ E(Γ), then i − j (mod n) ∈ S. Toida conjectured that if Γ0 is another circulant graph of order n, then Γ and Γ 0 are isomorphic if and only if they are isomorphic by a group automorphism of Zn. In this paper, we prove that Toida’s conjecture is true. We further prove that Toida’s conjecture implies Zibin’s conjecture, a generalization of Toida’s conjecture.
- ItemVertex-transitive digraphs with extra automorphisms that preserve the natural arc-colouring(The University of Queensland, Centre for Discrete Mathematics and Computing, 2017) Dobson, Ted; Hujdurovic, Ademir; Kutnar, Klavdija; Morris, JoyIn a Cayley digraph on a group G, if a distinct colour is assigned to each arc-orbit under the left-regular action of G, it is not hard to show that the elements of the left-regular action of G are the only digraph automorphisms that preserve this colouring. In this paper, we show that the equivalent statement is not true in the most straightforward generalisation to G-vertex-transitive digraphs, even if we restrict the situation to avoid some obvious potential problems. Speciﬁcally, we display an inﬁnite family of 2-closed groups G, and a G-arc-transitive digraph on each (without any digons) for which there exists an automorphism of the digraph that is not an element of G (it is an automorphism of G). Since the digraph is G-arc-transitive, the arcs would all be assigned the same colour under the colouring by arc-orbits, so this digraph automorphism is colour-preserving.