443 research outputs found

    On vertex-uniprimitive non-Cayley graphs of order pq

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    Let pp and qq be distinct odd primes. Let Γ=(V(Γ),E(Γ))\Gamma=(V(\Gamma), E(\Gamma)) be a non-Cayley vertex-transitive graph of order pq.pq. Let G\leq \Aut(\Gamma) acts primitively on the vertex set V(Γ)V(\Gamma). In this paper, we show that GG is uniprimitive which is primitive but not 2-transitive and we obtain some information about p,qp, q and the minimality of the Socle T=\soc(G).Comment: 5 page

    Cayley numbers with arbitrarily many distinct prime factors

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    A positive integer nn is a Cayley number if every vertex-transitive graph of order nn is a Cayley graph. In 1983, Dragan Maru\v{s}i\v{c} posed the problem of determining the Cayley numbers. In this paper we give an infinite set SS of primes such that every finite product of distinct elements from SS is a Cayley number. This answers a 1996 outstanding question of Brendan McKay and Cheryl Praeger, which they "believe to be the key unresolved question" on Cayley numbers. We also show that, for every finite product nn of distinct elements from SS, every transitive group of degree nn contains a semiregular element

    Cubic vertex-transitive non-Cayley graphs of order 12p

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    A graph is said to be {\em vertex-transitive non-Cayley} if its full automorphism group acts transitively on its vertices and contains no subgroups acting regularly on its vertices. In this paper, a complete classification of cubic vertex-transitive non-Cayley graphs of order 12p12p, where pp is a prime, is given. As a result, there are 1111 sporadic and one infinite family of such graphs, of which the sporadic ones occur when p=5p=5, 77 or 1717, and the infinite family exists if and only if p≡1 (mod  4)p\equiv1\ (\mod 4), and in this family there is a unique graph for a given order.Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematic

    Eigenvalues of Cayley graphs

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    We survey some of the known results on eigenvalues of Cayley graphs and their applications, together with related results on eigenvalues of Cayley digraphs and generalizations of Cayley graphs

    On groups all of whose Haar graphs are Cayley graphs

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    A Cayley graph of a group HH is a finite simple graph Γ\Gamma such that Aut(Γ){\rm Aut}(\Gamma) contains a subgroup isomorphic to HH acting regularly on V(Γ)V(\Gamma), while a Haar graph of HH is a finite simple bipartite graph Σ\Sigma such that Aut(Σ){\rm Aut}(\Sigma) contains a subgroup isomorphic to HH acting semiregularly on V(Σ)V(\Sigma) and the HH-orbits are equal to the bipartite sets of Σ\Sigma. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that the groups D6, D8, D10D_6, \, D_8, \, D_{10} and Q8Q_8 are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs (a group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian). As an application, it is also shown that every non-solvable group has a Haar graph which is not a Cayley graph.Comment: 17 page

    A classification of tetravalent edge-transitive metacirculants of odd order

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    In this paper a classification of tetravalent edge-transitive metacirculants is given. It is shown that a tetravalent edge-transitive metacirculant Γ\Gamma is a normal graph except for four known graphs. If further, Γ\Gamma is a Cayley graph of a non-abelian metacyclic group, then Γ\Gamma is half-transitive

    Cayley graphs of diameter two with order greater than 0.684 of the Moore bound for any degree

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    It is known that the number of vertices of a graph of diameter two cannot exceed d2+1d^2+1. In this contribution we give a new lower bound for orders of Cayley graphs of diameter two in the form C(d,2)>0.684d2C(d,2)>0.684d^2 valid for all degrees d≥360756d\geq 360756. The result is a significant improvement of currently known results on the orders of Cayley graphs of diameter two.Comment: 14 pages, 2 tables, Published in European Journal of Combinatorics. Free access to the article valid until July 9, 2016: http://authors.elsevier.com/a/1T3zuiVNjvDA

    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

    On Color Preserving Automorphisms of Cayley Graphs of Odd Square-free Order

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    An automorphism α\alpha of a Cayley graph Cay(G,S)Cay(G,S) of a group GG with connection set SS is color-preserving if α(g,gs)=(h,hs)\alpha(g,gs) = (h,hs) or (h,hs−1)(h,hs^{-1}) for every edge (g,gs)∈E(Cay(G,S))(g,gs)\in E(Cay(G,S)). If every color-preserving automorphism of Cay(G,S)Cay(G,S) is also affine, then Cay(G,S)Cay(G,S) is a CCA (Cayley color automorphism) graph. If every Cayley graph Cay(G,S)Cay(G,S) is a CCA graph, then GG is a CCA group. Hujdurovi\'c, Kutnar, D.W. Morris, and J. Morris have shown that every non-CCA group GG contains a section isomorphic to the nonabelian group F21F_{21} of order 2121. We first show that there is a unique non-CCA Cayley graph Γ\Gamma of F21F_{21}. We then show that if Cay(G,S)Cay(G,S) is a non-CCA graph of a group GG of odd square-free order, then G=H×F21G = H\times F_{21} for some CCA group HH, and Cay(G,S)=Cay(G,T)□ΓCay(G,S) = Cay(G,T)\Box\Gamma.Comment: 16 page

    Groups for which it is easy to detect graphical regular representations

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    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 phi of G such that phi(S) = S. We show that every nilpotent DRR-detecting group is a p-group, but that the wreath product of two cyclic groups of order p is not DRR-detecting, for every odd prime p. We also show that if G and H are nontrivial groups that admit a digraphical regular representation and either gcd(|G|,|H|) = 1, or H is not DRR-detecting, then the direct product G x H is not DRR-detecting. Some of these results also have analogues for graphical regular representations.Comment: 11 pages. v2: added acknowledgments and author addres
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