228 research outputs found

    Every finite group has a normal bi-Cayley graph

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    A graph \G with a group HH of automorphisms acting semiregularly on the vertices with two orbits is called a {\em bi-Cayley graph} over HH. When HH is a normal subgroup of \Aut(\G), we say that \G is {\em normal} with respect to HH. In this paper, we show that every finite group has a connected normal bi-Cayley graph. This improves Theorem~5 of [M. Arezoomand, B. Taeri, Normality of 2-Cayley digraphs, Discrete Math. 338 (2015) 41--47], and provides a positive answer to the Question of the above paper

    On congruence in Z^n and the dimension of a multidimensional circulant

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    From a generalization to ZnZ^n of the concept of congruence we define a family of regular digraphs or graphs called multidimensional circulants, which turn out to be Cayley (di)graphs of Abelian groups. This paper is mainly devoted to show the relationship between the Smith normal form for integral matrices and the dimensions of such (di)graphs, that is the minimum ranks of the groups they can arise from. In particular, those 2-step multidimensional circulants which are circulants, that is Cayley (di)graphs of cyclic groups, are fully characterized. In addition, a reasoning due to Lawrence is used to prove that the cartesian product of nn circulants with equal number of vertices p>2p>2, pp a prime, has dimension nn

    Derangement action digraphs and graphs

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    We study the family of \emph{derangement action digraphs}, which are a subfamily of the group action graphs introduced in [Fred Annexstein, Marc Baumslag, and Arnold L. Rosenberg, Group action graphs and parallel architectures, \emph{SIAM J. Comput.} 19 (1990), no. 3, 544--569]. For any non-empty set XX and a non-empty subset SS of \Der(X), the set of derangments of XX, we define the derangement action digraph DAβ†’(X;S)\rm\overrightarrow{DA}(X;S) to have vertex set XX, and an arc from xx to yy if and only if y=xsy=x^s for some s∈Ss\in S. In common with Cayley graphs and digraphs, derangement action digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on SS under which DAβ†’(X;S)\rm\overrightarrow{DA}(X;S) may be viewed as a simple graph of valency ∣S∣|S|, and we call such graphs derangement action graphs. Also we investigate the structural and symmetry properties of these digraphs and graphs. Several open problems are posed and many examples are given.Comment: 15 pages, 1 figur

    On Isomorphisms of Vertex-transitive Graphs

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    The isomorphism problem of Cayley graphs has been well studied in the literature, such as characterizations of CI (DCI)-graphs and CI (DCI)-groups. In this paper, we generalize these to vertex-transitive graphs and establish parallel results. Some interesting vertex-transitive graphs are given, including a first example of connected symmetric non-Cayley non-GI-graph. Also, we initiate the study for GI and DGI-groups, defined analogously to the concept of CI and DCI-groups

    Automorphism groups of circulant graphs -- a survey

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    A circulant (di)graph is a (di)graph on n vertices that admits a cyclic automorphism of order n. This paper provides a survey of the work that has been done on finding the automorphism groups of circulant (di)graphs, including the generalisation in which the edges of the (di)graph have been assigned colours that are invariant under the aforementioned cyclic automorphism.Comment: 16 pages, 0 figures, LaTeX fil

    A Generalisation of Isomorphisms with Applications

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    In this paper, we study the behaviour of TF-isomorphisms, a natural generalisation of isomorphisms. TF-isomorphisms allow us to simplify the approach to seemingly unrelated problems. In particular, we mention the Neighbourhood Reconstruction problem, the Matrix Symmetrization problem and Stability of Graphs. We start with a study of invariance under TF-isomorphisms. In particular, we show that alternating trails and incidence double covers are conserved by TF-isomorphisms, irrespective of whether they are TF-isomorphisms between graphs or digraphs. We then define an equivalence relation and subsequently relate its equivalence classes to the incidence double cover of a graph. By directing the edges of an incidence double cover from one colour class to the other and discarding isolated vertices we obtain an invariant under TF-isomorphisms which gathers a number of invariants. This can be used to study TF-orbitals, an analogous generalisation of the orbitals of a permutation group.Comment: 27 pages, 8 figure

    Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups

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    Let Ξ“=Cay(G,S)\Gamma=\mathrm{Cay}(G,S) be a Cayley digraph on a group GG and let A=Aut(Ξ“)A=\mathrm{Aut}(\Gamma). The Cayley index of Ξ“\Gamma is ∣A:G∣|A:G|. It has previously been shown that, if pp is a prime, GG is a cyclic pp-group and AA contains a noncyclic regular subgroup, then the Cayley index of Ξ“\Gamma is superexponential in pp. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if pp is an odd prime and GG is abelian but not cyclic, and has order a power of pp at least p3p^3, then there is a Cayley digraph Ξ“\Gamma on GG whose Cayley index is just pp, and whose automorphism group contains a nonabelian regular subgroup

    On the Normality of Some Cayley Digraphs with Valency 2

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    We call a Cayley digraph =Cay(G; S) normal for G if R(G), the rightregular representation of G, is a normal subgroup of the full automorphism groupAut() of . In this paper we determine the normality of Cayley digraphs of valency2 on the groups of order pq and also on non-abelian nite groups G such that everyproper subgroup of G is abelian.DOI : http://dx.doi.org/10.22342/jims.17.2.3.67-7

    Automorphism group of the complete alternating group graph

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    Let SnS_n and AnA_n denote the symmetric group and alternating group of degree nn with nβ‰₯3n\geq 3, respectively. Let SS be the set of all 33-cycles in SnS_n. The \emph{complete alternating group graph}, denoted by CAGnCAG_n, is defined as the Cayley graph Cay(An,S)\mathrm{Cay}(A_n,S) on AnA_n with respect to SS. In this paper, we show that CAGnCAG_n (nβ‰₯4n\geq 4) is not a normal Cayley graph. Furthermore, the automorphism group of CAGnCAG_n for nβ‰₯5n\geq 5 is obtained, which equals to Aut(CAGn)=(R(An)β‹ŠInn(Sn))β‹ŠZ2β‰…(Anβ‹ŠSn)β‹ŠZ2\mathrm{Aut}(CAG_n)=(R(A_n)\rtimes \mathrm{Inn}(S_n))\rtimes \mathbb{Z}_2\cong (A_n\rtimes S_n)\rtimes \mathbb{Z}_2, where R(An)R(A_n) is the right regular representation of AnA_n, Inn(Sn)\mathrm{Inn}(S_n) is the inner automorphism group of SnS_n, and Z2=⟨h⟩\mathbb{Z}_2=\langle h\rangle, where hh is the map Ξ±β†¦Ξ±βˆ’1\alpha\mapsto\alpha^{-1} (βˆ€Ξ±βˆˆAn\forall \alpha\in A_n).Comment: 9 pages, 1 figur

    Isomorphisms of Cayley graphs on nilpotent groups

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    Let S be a finite generating set of a torsion-free, nilpotent group G. We show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with multiplication by an element of the group.) More generally, we show that if Cay(G;S) and Cay(G';S') are connected Cayley graphs of finite valency on two nilpotent groups G and G', then every isomorphism from Cay(G;S) to Cay(G';S') factors through to a well-defined affine map from G/N to G'/N', where N and N' are the torsion subgroups of G and G', respectively. For the special case where the groups are abelian, these results were previously proved by A.A.Ryabchenko and C.Loeh, respectively.Comment: 12 pages, plus 7 pages of notes to aid the referee. One of our corollaries is already known, so a reference to the literature has been adde
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