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On a representation of the automorphism group of a graph in a unimodular group. / Estélyi, István; Karabáš, Ján; Nedela, Roman и др.
в: Discrete Mathematics, Том 344, № 12, 112606, 12.2021.Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
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TY - JOUR
T1 - On a representation of the automorphism group of a graph in a unimodular group
AU - Estélyi, István
AU - Karabáš, Ján
AU - Nedela, Roman
AU - Mednykh, Alexander
N1 - Funding Information: The first three authors were supported by the grant GACR 20-15576S . The first author acknowledges the financial support of Széchenyi 2020 under the EFOP-3.6.1-16-2016-00015 grant. The second and third author were supported by the grant No. APVV-19-0308 of Slovak Research and Development Agency . The fourth author was supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation . Funding Information: The authors express thanks to the anonymous referee for his/her useful comments which helped a lot to improve the presentation. The first three authors were supported by the grant GACR 20-15576S. The first author acknowledges the financial support of Sz?chenyi 2020 under the EFOP-3.6.1-16-2016-00015 grant. The second and third author were supported by the grant No. APVV-19-0308 of Slovak Research and Development Agency. The fourth author was supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation. Publisher Copyright: © 2021 Elsevier B.V.
PY - 2021/12
Y1 - 2021/12
N2 - We investigate a representation of the automorphism group of a connected graph X in the group of unimodular matrices Uβ of dimension β, where β is the Betti number of graph X. We classify the graphs for which the automorphism group does not embed into Uβ. It follows that if X has no pendant vertices and X is not a simple cycle, then the representation is faithful and AutX acts faithfully on H1(X,Z). The latter statement can be viewed as a discrete analogue of a classical Hurwitz's theorem on Riemann surfaces of genera greater than one.
AB - We investigate a representation of the automorphism group of a connected graph X in the group of unimodular matrices Uβ of dimension β, where β is the Betti number of graph X. We classify the graphs for which the automorphism group does not embed into Uβ. It follows that if X has no pendant vertices and X is not a simple cycle, then the representation is faithful and AutX acts faithfully on H1(X,Z). The latter statement can be viewed as a discrete analogue of a classical Hurwitz's theorem on Riemann surfaces of genera greater than one.
KW - Automorphism
KW - Graph
KW - Unimodular matrix
UR - http://www.scopus.com/inward/record.url?scp=85114015564&partnerID=8YFLogxK
U2 - 10.1016/j.disc.2021.112606
DO - 10.1016/j.disc.2021.112606
M3 - Article
AN - SCOPUS:85114015564
VL - 344
JO - Discrete Mathematics
JF - Discrete Mathematics
SN - 0012-365X
IS - 12
M1 - 112606
ER -
ID: 34154673