ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.06
https://doi.org/10.26493/2590-9770.1391.f46
(Also available at http://adam-journal.eu)
On automorphisms of Haar graphs
of abelian groups
Ted Dobson*
FAMNIT and IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia
Received 21 October 2020, accepted 16 February 2022, published online 18 July 2022
Abstract
Let G be a group and S ⊆ G. In this paper, a Haar graph of G with connection set S
has vertex set Z2 × G and edge set {(0, g)(1, gs) : g ∈ G and s ∈ S}. Haar graphs are
then natural bipartite analogues of Cayley digraphs, and are also called BiCayley graphs.
We first examine the relationship between the automorphism group of the Cayley digraph
of G with connection set S and the Haar graph of G with connection set S. We establish
that the automorphism group of a Haar graph contains a natural subgroup isomorphic to the
automorphism group of the corresponding Cayley digraph. In the case where G is abelian,
we show there are exactly four situations in which the automorphism group of the Haar
graph can be larger than the natural subgroup corresponding to the automorphism group
of the Cayley digraph together with a specific involution, and analyze the full automor-
phism group in each of these cases. As an application, we show that all s-transitive Cayley
graphs of generalized dihedral groups have a quasiprimitive automorphism group, can be
constructed from digraphs of smaller order, or are Haar graphs of abelian groups whose
automorphism groups have a particular permutation group theoretic property.
Keywords: Groups, graphs.
Math. Subj. Class.: 05C15, 05C10
*The author is indebted to the referees for their careful reading of the paper, and their suggestions for improve-
ments.
E-mail address: ted.dobson@upr.si (Ted Dobson)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.06
https://doi.org/10.26493/2590-9770.1391.f46
(Dostopno tudi na http://adam-journal.eu)
Avtomorfizmi Haarovih grafov abelskih grup
Ted Dobson*
FAMNIT and IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia
Prejeto 21. oktobra 2020, sprejeto 16. februarja 2022, objavljeno na spletu 18. julija 2022
Povzetek
Naj bo G grupa in S ⊆ G. V tem članku je Haarov graf grupe G, ki pripada množici
povezav S, graf, katerega množica vozlišč je Z2×G, množica povezav pa je {(0, g)(1, gs) :
g ∈ G and s ∈ S}. Haarovi grafi so naravni dvodelni analogi Cayleyjevih digrafov;
imenujejo jih tudi biCayleyjevi grafi. Najprej pregledamo odnos med grupo avtomofiz-
mov Cayleyjevega digrafa grupe G s povezavno množico S in Haarovega grafa grupe
G s povezavno množico S. Ugotovimo, da grupa avtomorfizmov Haarovega grafa vse-
buje naravno podgrupo, izmorfno grupi avtomorfizmov ustreznega Cayleyjevega digrafa.
V primeru, da je grupa G abelska, pokažemo, da obstajajo natanko štiri situacije, v ka-
terih je grupa avtomorfizmov Haarovega grafa lahko večja od naravne podgrupe, ki ustreza
grupi avtomorfizmov Cayleyjevega digrafa skupaj s specifično involucijo; v vsakem od
teh primerov analiziramo polno grupo avtomorfizmov. Na osnovi tega pokažemo, da za
vse s-tranzitivne Cayleyjeve grafe posplošenih diedrskih grup velja, da imajo kvaziprimi-
tivno grupo avtomorfizmov, da se dajo konstruirati iz digrafov manjšega reda, ali pa gre za
Haarove grafe abelskih grup, katerih grupe avtomorfizmov imajo določeno posebno last-
nost permutacijskih grup.
Ključne besede: Grupe, grafi.
Math. Subj. Class.: 05C15, 05C10
*Avtor je hvaležen recenzentom za njihovo skrbno branje članka ter za njihove predloge izboljšav.
E-poštni naslov: ted.dobson@upr.si (Ted Dobson)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/