Tutte polynomials of vertex-weighted graphs and group cohomology
type: Item
instance of: scholarly article
title: Tutte polynomials of vertex-weighted graphs and group cohomology
zbMATH Open document ID: 1479.05162
DOI: 10.1134/S0040577921050056
author: A. A. Kazakov
author: Dmitry Talalaev
author: B. S. Bychkov
published in: Theoretical and Mathematical Physics
publication date: 2021-06-29T00:00:00Z
review text: The classical Tutte polynomial \(T_{G}(x,y)\) of a graph \(G\) initially appeared in the combinatorial theory of graphs as a generalization of a recurrence relation discovered by Whitney and Birkhoff for the number of proper colorings of a graph in \(n\) colors. The polynomial thus defined turns out to be related to many important graph invariants: the number of proper colorings in \(n\) colors, the number of \(n\)-flows, the number of spanning trees, the number of dimers (one-factors), etc.. Therefore, the theory of the Tutte polynomial was highly developed and generalized in many ways. The classical Tutte polynomial is discussed and defined recursively via deletion-contraction relations. The authors generalized the classical relation. Here, the authors constructed a generalization of the Tutte polynomial for vertex-weighted graphs for which the coefficients of the deletion-contraction relation depend nontrivially on the vertex weights. They showed that the corresponding relation on the coefficients coincides with the two-cocycle relation in the group cohomology. Also, they obtained a representation of a new invariant by summing over subgraphs and established its connection with four-invariants of graphs.
reviewed by: V. Lokesha
Mathematics Subject Classification ID: 05C31
Mathematics Subject Classification ID: 05C22
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 57M15
zbMATH DE Number: 7364290
zbMATH Keywords: Tutte polynomial
zbMATH Keywords: group cohomology
zbMATH Keywords: combinatorial graphs
zbMATH Keywords: vertex weighted graph
MaRDI profile type: Publication
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full work available at URL: s0040577921050056
OpenAlex ID: W3174068080
description: scientific article; zbMATH DE number 7364290