Interlace polynomials and Tutte polynomials

TL;DR

This paper explores the relationship between interlace polynomials of graphs and Tutte polynomials of associated binary matroids, providing a unified algebraic framework for understanding and deriving various interlace polynomials.

Contribution

It establishes a connection between interlace polynomials and parametrized Tutte polynomials of specific binary matroids, offering a new algebraic perspective.

Findings

Interlace polynomials can be obtained from Tutte polynomials of binary matroids.

Parametrized Tutte polynomials of (I | A(G)) and (I | A(G) | I+A(G)) represent various interlace polynomials.

Provides a unified algebraic framework for interlace and Tutte polynomials.

Abstract

Let G be a graph with adjacency matrix A(G). Consider the matrix IA(G)=(I | A(G)), where I is the identity matrix, and let M(IA(G)) be the binary matroid represented by IA(G). Then suitably parametrized versions of the Tutte polynomial of M(IA(G)) yield the interlace polynomials of G, introduced by Arratia, Bollob\'as and Sorkin [J. Combin. Theory Ser. B 92 (2004) 199-233; Combinatorica 24 (2004) 567-584]. Interlace polynomials subsequently introduced by other authors may be obtained from parametrized Tutte polynomials of the binary matroid represented by (I | A(G) | I+A(G)).