@article{d6c4ee9c475e4553956867ee6375341c,
title = "On a representation of the automorphism group of a graph in a unimodular group",
abstract = "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.",
keywords = "Automorphism, Graph, Unimodular matrix",
author = "Istv{\'a}n Est{\'e}lyi and J{\'a}n Karab{\'a}{\v s} and Roman Nedela and Alexander Mednykh",
note = "Funding Information: The first three authors were supported by the grant GACR 20-15576S . The first author acknowledges the financial support of Sz{\'e}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 . 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: {\textcopyright} 2021 Elsevier B.V.",
year = "2021",
month = dec,
doi = "10.1016/j.disc.2021.112606",
language = "English",
volume = "344",
journal = "Discrete Mathematics",
issn = "0012-365X",
publisher = "Elsevier",
number = "12",
}