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On automorphism groups of circulant digraphs of square-free order

Edward Dobson, Joy Morris
2005 Discrete Mathematics  
We show that the full automorphism group of a circulant digraph of square-free order is either the intersection of two 2-closed groups, each of which is the wreath product of 2-closed groups of smaller  ...  degree, or contains a transitive normal subgroup which is the direct product of two 2-closed groups of smaller degree.  ...  time algorithm to calculate the full automorphism group of a circulant digraph of square-free order can be derived using this result.  ... 
doi:10.1016/j.disc.2004.03.018 fatcat:7twplujkhvcm7kvb5wyujkv4o4

Automorphism Groups of Circulant Digraphs With Applications to Semigroup Theory

João Araújo, Wolfram Bentz, Edward Dobson, Janusz Konieczny, Joy Morris
2017 Combinatorica  
We characterize the automorphism groups of circulant digraphs whose connection sets are relatively small, and of unit circulant digraphs.  ...  Then we use these characterizations to prove results on the automorphisms of the endomorphism monoids of those digraphs.  ...  In the special case of circulant digraphs of square-free order n, a result equivalent to this result was proved independently in [18] .  ... 
doi:10.1007/s00493-016-3403-0 fatcat:vmmn4jq6xbbjjb24utw6mztij4

Asymptotic Automorphism Groups of Circulant Graphs and Digraphs [article]

Soumya Bhoumik, Edward Dobson, Joy Morris
2012 arXiv   pre-print
Of the circulant graphs that do not have automorphism group as small as possible, we give some families of integers such that it is not true that almost all circulant graphs whose order lies in any one  ...  It is then shown that there is a large family of integers for which almost every circulant digraph whose order lies in this family and that does not have automorphism group as small as possible, is normal  ...  In the special case of circulant digraphs of square-free order n, an equivalent result was proven independently in [5] . Theorem 2.5 Let G ≤ S n contain a regular cyclic subgroup ρ .  ... 
arXiv:1203.0707v1 fatcat:jgswnnxiabfgvccmbiu4e7pnqy

Automorphism groups of circulant graphs -- a survey [article]

Joy Morris
2004 arXiv   pre-print
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  ...  The result on circulant graphs on a square-free number of vertices is the culmination of two densely-packed, long papers; the first provides a structural theorem, and the second the algorithm.  ... 
arXiv:math/0411302v1 fatcat:2etilnajr5ga5ngpfxvdyjjhje

A class of vertex-transitive digraphs, II

Chong-Yun Chao, Jacqueline G Wells
1982 Journal of combinatorial theory. Series B (Print)  
The nonisomorphic classes of these circulant digraphs with pq vertices are enumerated. (2).  ...  We determine the number of non-isomorphic classes of self-complementary circulant digraphs with pq vertices, where p and q are distinct primes.  ...  Also, a digraph on n vertices is said to be a circulant digraph if its group of automorphisms contains the cyclic group of order n.  ... 
doi:10.1016/0095-8956(82)90009-0 fatcat:amji77nrxzaelndspixbe7l6p4

Page 7332 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews  
The authors determine the automorphism groups of circulant digraphs of outdegree 3.  ...  of circulant digraphs of degree 3.  ... 

Two families of graphs that are Cayley on nonisomorphic groups [article]

Joy Morris, Josip Smolcic
2020 arXiv   pre-print
A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group.  ...  The work to date has focussed on a few special situations: when the groups are p-groups; when the groups have order pq; when the Cayley graphs are normal; or when the groups are both abelian.  ...  Marušič and the first author studied the question of which normal circulant graphs of square-free order are also Cayley graphs of a nonabelian group [10] .  ... 
arXiv:2005.11585v1 fatcat:icp4k7vur5eyfbyed5fu4ot5xy

On the automorphism groups of almost all circulant graphs and digraphs

Soumya Bhoumik, Ted Dobson, Joy Morris
2014 Ars Mathematica Contemporanea  
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.  ...  every non-normal circulant digraph.  ...  Acknowledgement: The authors are indebted to the anonymous referees whose suggestions improved the clarity of the proofs as well as the exposition in this manuscript.  ... 
doi:10.26493/1855-3974.315.868 fatcat:bobfdfq6krgfxnmlh67ssrlo7y

Page 792 of Mathematical Reviews Vol. , Issue 90B [page]

1990 Mathematical Reviews  
Let Z, be the cyclic group of odd prime order p.  ...  (B) Minimize the value k of the diameter with a given order N of a circulant.  ... 

Enumeration of Cayley graphs and digraphs

Brian Alspach, Marni Mishna
2002 Discrete Mathematics  
Both the directed and undirected cases of the following three families are studied: Cayley graphs on cyclic groups, known as circulant graphs, of square-free order, circulant graphs of arbitrary order  ...  n whose connection sets are subsets of the group of units of Zn, and Cayley graphs on elementary abelian groups.  ...  Circulant graphs and digraphs Turner's result applies to Cayley graphs on cyclic groups of prime order.  ... 
doi:10.1016/s0012-365x(02)00319-9 fatcat:dstva3tdffeqrf35gjfg7cjlei

Two families of graphs that are Cayley on nonisomorphic groups

Joy MORRİS, Josip SMOLCİC
2021 Journal of Algebra Combinatorics Discrete Structures and Applications  
A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group.  ...  The work to date has focussed on a few special situations: when the groups are p-groups; when the groups have order pq; when the Cayley graphs are normal; or when the groups are both abelian.  ...  Acknowledgment: The authors thank the anonymous referees for helpful comments on this paper.  ... 
doi:10.13069/jacodesmath.867644 fatcat:qdmamp7t25c5vlhyk3lyeai7pi

Contents

2005 Discrete Mathematics  
Dobson On groups of odd prime-power degree that contain a full cycle 65 E. Dobson and J. Morris On automorphism groups of circulant digraphs of square-free order 79 D. Froncek and B.  ...  Brunat On endo-Cayley digraphs: The hamiltonian property 194 D.W. Morris, J. Morris and D.P. Moulton Flows that are sums of hamiltonian cycles in Cayley graphs on abelian groups 208 J.  ... 
doi:10.1016/s0012-365x(05)00407-3 fatcat:rxvtxpybhzgy3krwwreznodnvy

Page 8087 of Mathematical Reviews Vol. , Issue 2003k [page]

2003 Mathematical Reviews  
Muzychuk [Discrete Math. 176 (1997), no. 1-3, 285-298; MR 98h:05141b], all circulant digraphs of order n are CI if and only if n is square-free or n = 4m, where m is odd square-free.  ...  An almost identical result is valid for undirected circulant graphs. The automorphism group of (Z,,+) can be identified with the multiplicative group Z* of invertible elements (units) of Z,,.  ... 

A solution of an equivalence problem for semisimple cyclic codes [article]

M.Muzychuk
2011 arXiv   pre-print
In this paper we propose an efficient solution of an equivalence problem for semisimple cyclic codes.  ...  It was shown in [12] [13] that a cyclic group of a square-free or twice square free order is a CI-group with respect to colored Cayley digraphs.  ...  A cyclic group of a square-free or twice square-free order is a CI-group with respect to semisimple cyclic codes.  ... 
arXiv:1105.4320v1 fatcat:ujho2d6chbex7h4opvs4ujs5ra

On Solvable Groups and Circulant Graphs

Edward Dobson
2000 European journal of combinatorics (Print)  
We prove that a vertex-transitive graph of order n, with gcd(n, ϕ(n)) = 1, is isomorphic to a circulant graph of order n if and only if Aut( ) contains a transitive solvable subgroup.  ...  As a corollary, we prove that every vertex-transitive graph of order n is isomorphic to a circulant graph of order n if and only if for every such , Aut( ) contains a transitive solvable subgroup and n  ...  A circulant graph of order n is a graph whose automorphism group contains an n-cycle.  ... 
doi:10.1006/eujc.2000.0412 fatcat:afn2jutfkjgqrectl3vuy4wgwy
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