Tensor product formulas for the Bollob\'as-Riordan and Krushkal polynomials

TL;DR

This paper generalizes tensor product formulas for various graph polynomials, including Bollobás-Riordan and Krushkal polynomials, by defining tensor products of embedded graphs in pseudo-surfaces, unifying previous results.

Contribution

It introduces a unified framework for tensor product formulas applicable to Bollobás-Riordan and Krushkal polynomials for graphs embedded in pseudo-surfaces.

Findings

Derived Brylawski-style tensor product formulas for Bollobás-Riordan and Krushkal polynomials.

Unified previous disparate tensor product formulas under a common framework.

Extended tensor product concepts to graphs embedded in pseudo-surfaces.

Abstract

Brylawski's tensor product formula expresses the Tutte polynomial of the tensor product of two graphs in terms of Tutte polynomials arising from the tensor factors. Analogous tensor product formulas are known for the ribbon graph polynomial and transition polynomials of graphs embedded in surfaces, as well as for the Bollob\'as-Riordan polynomial in some special cases. We define the tensor product of graphs embedded in pseudo-surfaces and use this to generalize and unify all of the above results, providing Brylawski-style formulas for both the Bollob\'as-Riordan and Krushkal polynomials.