On the evaluation of the Tutte polynomial at the points (1,-1) and (2,-1)
International audience C. Merino [Electron. J. Combin. 15 (2008)] showed that the Tutte polynomial of a complete graph satisfies $t(K_{n+2};2,-1)=t(K_n;1,-1)$. We first give a bijective proof of this identity based on the relationship between the Tutte polynomial and the inversion polynomial for tre...
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ftccsdartic:oai:HAL:hal-01215102v1 2023-05-15T16:51:29+02:00 On the evaluation of the Tutte polynomial at the points (1,-1) and (2,-1) Goodall, Andrew Merino, Criel de Mier, Anna Noy, Marc Department of Applied Mathematics and Institute of Theoretical Computer Science (Charles University) Charles University Prague (CU) Instituto de Matematicas (UNAM) Universidad Nacional Autónoma de México (UNAM) Universitat Politècnica de Catalunya Barcelona (UPC) Bousquet-Mélou Mireille and Wachs Michelle and Hultman Axel Reykjavik, Iceland 2011 https://hal.inria.fr/hal-01215102 https://hal.inria.fr/hal-01215102/document https://hal.inria.fr/hal-01215102/file/dmAO0137.pdf en eng HAL CCSD Discrete Mathematics and Theoretical Computer Science DMTCS hal-01215102 https://hal.inria.fr/hal-01215102 https://hal.inria.fr/hal-01215102/document https://hal.inria.fr/hal-01215102/file/dmAO0137.pdf info:eu-repo/semantics/OpenAccess ISSN: 1462-7264 EISSN: 1365-8050 Discrete Mathematics and Theoretical Computer Science 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) https://hal.inria.fr/hal-01215102 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.411-422 generating function up-down permutation Tutte polynomial increasing tree threshold graph [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] info:eu-repo/semantics/conferenceObject Conference papers 2011 ftccsdartic 2021-11-14T00:40:05Z International audience C. Merino [Electron. J. Combin. 15 (2008)] showed that the Tutte polynomial of a complete graph satisfies $t(K_{n+2};2,-1)=t(K_n;1,-1)$. We first give a bijective proof of this identity based on the relationship between the Tutte polynomial and the inversion polynomial for trees. Next we move to our main result, a sufficient condition for a graph G to have two vertices u and v such that $t(G;2,-1)=t(G-\{u,v\};1,-1)$; the condition is satisfied in particular by the class of threshold graphs. Finally, we give a formula for the evaluation of $t(K_{n,m};2,-1)$ involving up-down permutations. C. Merino [Electron. J. Combin. 15 (2008)] a montré que le polynôme de Tutte du graphe complet satisfait $t(K_{n+2};2,-1)=t(K_n;1,-1)$. Le rapport entre le polynôme de Tutte et le polynôme d'inversions d'un arbre nous permet de donner une preuve bijective de cette identité. Le résultat principal du travail est une condition suffisante pour qu'un graphe ait deux sommets u et v tels que $t(G;2,-1)=t(G-\{u,v\};1,-1)$; en particulier, les graphes ``threshold'' satisfont cette condition. Finalement, nous donnons une formule pour $t(K_{n,m};2,-1)$ qui fait intervenir les permutations alternées. Conference Object Iceland Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
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Open Polar |
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Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
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ftccsdartic |
language |
English |
topic |
generating function up-down permutation Tutte polynomial increasing tree threshold graph [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
spellingShingle |
generating function up-down permutation Tutte polynomial increasing tree threshold graph [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] Goodall, Andrew Merino, Criel de Mier, Anna Noy, Marc On the evaluation of the Tutte polynomial at the points (1,-1) and (2,-1) |
topic_facet |
generating function up-down permutation Tutte polynomial increasing tree threshold graph [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
description |
International audience C. Merino [Electron. J. Combin. 15 (2008)] showed that the Tutte polynomial of a complete graph satisfies $t(K_{n+2};2,-1)=t(K_n;1,-1)$. We first give a bijective proof of this identity based on the relationship between the Tutte polynomial and the inversion polynomial for trees. Next we move to our main result, a sufficient condition for a graph G to have two vertices u and v such that $t(G;2,-1)=t(G-\{u,v\};1,-1)$; the condition is satisfied in particular by the class of threshold graphs. Finally, we give a formula for the evaluation of $t(K_{n,m};2,-1)$ involving up-down permutations. C. Merino [Electron. J. Combin. 15 (2008)] a montré que le polynôme de Tutte du graphe complet satisfait $t(K_{n+2};2,-1)=t(K_n;1,-1)$. Le rapport entre le polynôme de Tutte et le polynôme d'inversions d'un arbre nous permet de donner une preuve bijective de cette identité. Le résultat principal du travail est une condition suffisante pour qu'un graphe ait deux sommets u et v tels que $t(G;2,-1)=t(G-\{u,v\};1,-1)$; en particulier, les graphes ``threshold'' satisfont cette condition. Finalement, nous donnons une formule pour $t(K_{n,m};2,-1)$ qui fait intervenir les permutations alternées. |
author2 |
Department of Applied Mathematics and Institute of Theoretical Computer Science (Charles University) Charles University Prague (CU) Instituto de Matematicas (UNAM) Universidad Nacional Autónoma de México (UNAM) Universitat Politècnica de Catalunya Barcelona (UPC) Bousquet-Mélou Mireille and Wachs Michelle and Hultman Axel |
format |
Conference Object |
author |
Goodall, Andrew Merino, Criel de Mier, Anna Noy, Marc |
author_facet |
Goodall, Andrew Merino, Criel de Mier, Anna Noy, Marc |
author_sort |
Goodall, Andrew |
title |
On the evaluation of the Tutte polynomial at the points (1,-1) and (2,-1) |
title_short |
On the evaluation of the Tutte polynomial at the points (1,-1) and (2,-1) |
title_full |
On the evaluation of the Tutte polynomial at the points (1,-1) and (2,-1) |
title_fullStr |
On the evaluation of the Tutte polynomial at the points (1,-1) and (2,-1) |
title_full_unstemmed |
On the evaluation of the Tutte polynomial at the points (1,-1) and (2,-1) |
title_sort |
on the evaluation of the tutte polynomial at the points (1,-1) and (2,-1) |
publisher |
HAL CCSD |
publishDate |
2011 |
url |
https://hal.inria.fr/hal-01215102 https://hal.inria.fr/hal-01215102/document https://hal.inria.fr/hal-01215102/file/dmAO0137.pdf |
op_coverage |
Reykjavik, Iceland |
genre |
Iceland |
genre_facet |
Iceland |
op_source |
ISSN: 1462-7264 EISSN: 1365-8050 Discrete Mathematics and Theoretical Computer Science 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) https://hal.inria.fr/hal-01215102 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.411-422 |
op_relation |
hal-01215102 https://hal.inria.fr/hal-01215102 https://hal.inria.fr/hal-01215102/document https://hal.inria.fr/hal-01215102/file/dmAO0137.pdf |
op_rights |
info:eu-repo/semantics/OpenAccess |
_version_ |
1766041614506000384 |