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Conference Papers Discrete Mathematics and Theoretical Computer Science Year : 2011

## On the evaluation of the Tutte polynomial at the points (1,-1) and (2,-1)

Criel Merino
• Function : Author
Anna de Mier
• Function : Author
Marc Noy
• Function : Author
• PersonId : 843456

#### Abstract

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.

### Dates and versions

hal-01215102 , version 1 (13-10-2015)

### Identifiers

• HAL Id : hal-01215102 , version 1
• DOI :

### Cite

Andrew Goodall, Criel Merino, Anna de Mier, Marc Noy. On the evaluation of the Tutte polynomial at the points (1,-1) and (2,-1). 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.411-422, ⟨10.46298/dmtcs.2921⟩. ⟨hal-01215102⟩

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