The Jacobian of a graph and graph automorphisms
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2022
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929425450270720 |
|---|---|
| author | Estélyi, István Karabáš, Ján Mednykh, Alexander Nedela, Roman |
| author_facet | Estélyi, István Karabáš, Ján Mednykh, Alexander Nedela, Roman |
| contents | 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 automorphims, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_01469 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Jacobian of a graph and graph automorphisms Estélyi, István Karabáš, Ján Mednykh, Alexander Nedela, Roman Combinatorics Algebraic Geometry Group Theory 05C50, 05C21, 20B25 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 automorphims, 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. |
| title | The Jacobian of a graph and graph automorphisms |
| topic | Combinatorics Algebraic Geometry Group Theory 05C50, 05C21, 20B25 |
| url | https://arxiv.org/abs/2206.01469 |