The asymptotic spectrum of graphs and the Shannon capacity

TL;DR

This paper introduces the asymptotic spectrum of graphs and uses it to provide a new dual characterization of the Shannon capacity, involving known graph parameters like the Lovász theta number.

Contribution

It extends the theory of asymptotic spectra to graphs, offering a novel dual perspective on Shannon capacity using spectral graph parameters.

Findings

New dual characterization of Shannon capacity

Includes known graph parameters in the spectrum

Provides a framework for analyzing graph capacities

Abstract

We introduce the asymptotic spectrum of graphs and apply the theory of asymptotic spectra of Strassen (J. Reine Angew. Math. 1988) to obtain a new dual characterisation of the Shannon capacity of graphs. Elements in the asymptotic spectrum of graphs include the Lov\'asz theta number, the fractional clique cover number, the complement of the fractional orthogonal rank and the fractional Haemers bounds.