The Jacobian of a graph and graph automorphisms

The Jacobian of a graph and graph automorphisms Full article

Journal

Discrete Mathematics
ISSN: 0012-365X , E-ISSN: 1872-681X

Output data

Year: 2025, Volume: 348, Number: 2, Article number : 114259, Pages count : 8 DOI: 10.1016/j.disc.2024.114259

Tags

Graph, Flow, Automorphism, Jacobian

Authors

Estélyi István ¹ , Karabáš Ján ⁵ , Mednykh Alexander ^2,3 , Nedela Roman ^4,5

Affiliations

1	Faculty of Information Technology, University of Pannonia
2	Sobolev Institute of Mathematics
3	Novosibirsk State University
4	Faculty of Applied Sciences, University of West Bohemia
5	Mathematical Institute of Slovak Academy of Sciences

Funding (1)

Sobolev Institute of Mathematics

FWNF-2022-0005

In the present paper we investigate the faithfulness of certain linear representations of groups of automorphisms of a graph X in the group of symmetries of the Jacobian of X. As a consequence we show that if a 3-edge-connected graph X admits a nonabelian semiregular group of automorphisms, then the Jacobian of X cannot be cyclic. In particular, Cayley graphs of degree at least three arising from nonabelian groups have non-cyclic Jacobians. While the size of the Jacobian of X is well-understood – it is equal to the number of spanning trees of X– the combinatorial interpretation of the rank of Jacobian of a graph is unknown. Our paper presents a contribution in this direction.

Estélyi I. , Karabáš J. , Mednykh A. , Nedela R.
The Jacobian of a graph and graph automorphisms
Discrete Mathematics. 2025. V.348. N2. 114259 :1-8. DOI: 10.1016/j.disc.2024.114259 WOS Scopus OpenAlex

Submitted:	Dec 22, 2022
Accepted:	Sep 4, 2024
Published online:	Sep 11, 2024
Published print:	Feb 5, 2025

Web of science:	WOS:001315207200001
Scopus:	2-s2.0-85203432938
OpenAlex:	W4402448510

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