Duality of Graph Invariants

TL;DR

This paper introduces a new duality framework for graph invariants using a non-linear transform, revealing fixed points among key invariants and exploring implications for quantum non-locality.

Contribution

It defines a novel non-linear transform on graph invariants and uncovers duality relations, including fixed points for several important invariants.

Findings

Weighted independence number is a fixed point of the transform squared.

Weighted Lovász number is a fixed point of the transform squared.

Weighted Shannon capacity is not a fixed point.

Abstract

We study a new set of duality relations between weighted, combinatoric invariants of a graph $G$. The dualities arise from a non-linear transform $\mathfrak{B}$, acting on the weight function $p$. We define $\mathfrak{B}$ on a space of real-valued functions $\mathcal{O}$ and investigate its properties. We show that three invariants (weighted independence number, weighted Lov\'{a}sz number, and weighted fractional packing number) are fixed points of $\mathfrak{B}^{2}$, but the weighted Shannon capacity is not. We interpret these invariants in the study of quantum non-locality.