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On automorphism groups of circulant digraphs of square-free order

2005
*
Discrete Mathematics
*

We show that the full

doi:10.1016/j.disc.2004.03.018
fatcat:7twplujkhvcm7kvb5wyujkv4o4
*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. ...##
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Automorphism Groups of Circulant Digraphs With Applications to Semigroup Theory

2017
*
Combinatorica
*

We characterize the

doi:10.1007/s00493-016-3403-0
fatcat:vmmn4jq6xbbjjb24utw6mztij4
*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] . ...##
###
Asymptotic Automorphism Groups of Circulant Graphs and Digraphs
[article]

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 . ...

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

2004
*
arXiv
*
pre-print

A

arXiv:math/0411302v1
fatcat:2etilnajr5ga5ngpfxvdyjjhje
*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. ...##
###
A class of vertex-transitive digraphs, II

1982
*
Journal of combinatorial theory. Series B (Print)
*

The nonisomorphic classes

doi:10.1016/0095-8956(82)90009-0
fatcat:amji77nrxzaelndspixbe7l6p4
*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. ...##
###
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]

2020
*
arXiv
*
pre-print

A number

arXiv:2005.11585v1
fatcat:icp4k7vur5eyfbyed5fu4ot5xy
*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] . ...##
###
On the automorphism groups of almost all circulant graphs and digraphs

2014
*
Ars Mathematica Contemporanea
*

We attempt to determine the structure

doi:10.26493/1855-3974.315.868
fatcat:bobfdfq6krgfxnmlh67ssrlo7y
*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. ...##
###
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

2002
*
Discrete Mathematics
*

Both the directed and undirected cases

doi:10.1016/s0012-365x(02)00319-9
fatcat:dstva3tdffeqrf35gjfg7cjlei
*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*. ...##
###
Two families of graphs that are Cayley on nonisomorphic groups

2021
*
Journal of Algebra Combinatorics Discrete Structures and Applications
*

A number

doi:10.13069/jacodesmath.867644
fatcat:qdmamp7t25c5vlhyk3lyeai7pi
*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. ...##
###
Contents

2005
*
Discrete Mathematics
*

Dobson

doi:10.1016/s0012-365x(05)00407-3
fatcat:rxvtxpybhzgy3krwwreznodnvy
*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. ...##
###
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]

2011
*
arXiv
*
pre-print

In this paper we propose an efficient solution

arXiv:1105.4320v1
fatcat:ujho2d6chbex7h4opvs4ujs5ra
*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. ...##
###
On Solvable Groups and Circulant Graphs

2000
*
European journal of combinatorics (Print)
*

We prove that a vertex-transitive graph

doi:10.1006/eujc.2000.0412
fatcat:afn2jutfkjgqrectl3vuy4wgwy
*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. ...
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