Interlace Polynomials: Enumeration, Unimodality, and Connections to Codes

TL;DR

This paper studies interlace polynomials of graphs, providing enumeration results up to 12 vertices, disproving a unimodality conjecture, and exploring connections to error-correcting codes and quantum states.

Contribution

It offers the first enumeration of interlace polynomials for graphs up to 12 vertices and links their properties to coding theory and quantum information.

Findings

Existence of non-unimodal interlace polynomials for graphs larger than 9 vertices.

All interlace polynomials of graphs up to 12 vertices are unimodal.

Disproof of the conjecture that all such polynomials are unimodal.

Abstract

The interlace polynomial q was introduced by Arratia, Bollobas, and Sorkin. It encodes many properties of the orbit of a graph under edge local complementation (ELC). The interlace polynomial Q, introduced by Aigner and van der Holst, similarly contains information about the orbit of a graph under local complementation (LC). We have previously classified LC and ELC orbits, and now give an enumeration of the corresponding interlace polynomials of all graphs of order up to 12. An enumeration of all circle graphs of order up to 12 is also given. We show that there exist graphs of all orders greater than 9 with interlace polynomials q whose coefficient sequences are non-unimodal, thereby disproving a conjecture by Arratia et al. We have verified that for graphs of order up to 12, all polynomials Q have unimodal coefficients. It has been shown that LC and ELC orbits of graphs correspond to equivalence classes of certain error-correcting codes and quantum states. We show that the properties of these codes and quantum states are related to properties of the associated interlace polynomials.