Tensor products of multimatroids and a Brylawski-type formula for the transition polynomial

Iain Moffatt; Maya Thompson; Steven Noble

Tensor products of multimatroids and a Brylawski-type formula for the transition polynomial

Iain Moffatt
, Maya Thompson
, Steven Noble

Research output: Contribution to journal › Article › peer-review

Abstract

Brylawski’s tensor product formula expresses the Tutte polynomial of the tensor
product of two graphs or matroids in terms of Tutte polynomials arising from the
tensor factors. Analogous tensor product formulas are known for the Bollobas–
Riordan and transition polynomials of graphs embedded in surfaces. We show
that these formulas are instances of a more general result for multimatroids by
giving a tensor product formula for the multimatroid transition polynomial and
showing that Brylawski’s formula and its topological analogues arise from it.
Along the way we provide formulas for the transition polynomial of the two-sum
and star product of multimatroids.

Original language	English
Journal	Annals of combinatorics
Publication status	Accepted/In press - 23 Jul 2026

Embargoed Document

Accepted Manuscript
Accepted author manuscript, 4.2 MB
Licence: CC BY

Cite this

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title = "Tensor products of multimatroids and a Brylawski-type formula for the transition polynomial",

abstract = "Brylawski{\textquoteright}s tensor product formula expresses the Tutte polynomial of the tensor product of two graphs or matroids in terms of Tutte polynomials arising from the tensor factors. Analogous tensor product formulas are known for the Bollobas{\textendash} Riordan and transition polynomials of graphs embedded in surfaces. We show that these formulas are instances of a more general result for multimatroids by giving a tensor product formula for the multimatroid transition polynomial and showing that Brylawski{\textquoteright}s formula and its topological analogues arise from it. Along the way we provide formulas for the transition polynomial of the two-sum and star product of multimatroids.",

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AU - Thompson, Maya

AU - Noble, Steven

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N2 - Brylawski’s tensor product formula expresses the Tutte polynomial of the tensor product of two graphs or matroids in terms of Tutte polynomials arising from the tensor factors. Analogous tensor product formulas are known for the Bollobas– Riordan and transition polynomials of graphs embedded in surfaces. We show that these formulas are instances of a more general result for multimatroids by giving a tensor product formula for the multimatroid transition polynomial and showing that Brylawski’s formula and its topological analogues arise from it. Along the way we provide formulas for the transition polynomial of the two-sum and star product of multimatroids.

AB - Brylawski’s tensor product formula expresses the Tutte polynomial of the tensor product of two graphs or matroids in terms of Tutte polynomials arising from the tensor factors. Analogous tensor product formulas are known for the Bollobas– Riordan and transition polynomials of graphs embedded in surfaces. We show that these formulas are instances of a more general result for multimatroids by giving a tensor product formula for the multimatroid transition polynomial and showing that Brylawski’s formula and its topological analogues arise from it. Along the way we provide formulas for the transition polynomial of the two-sum and star product of multimatroids.

M3 - Article

SN - 0218-0006

JO - Annals of combinatorics

JF - Annals of combinatorics

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