A variant of the Lov\'asz-Theta number based on projection matrices

TL;DR

This paper introduces a new graph parameter called the projection theta number, based on combinatorial projection matrices, which relates to existing Lovász theta variants and offers computational advantages for certain graphs.

Contribution

The paper defines a novel SDP-based graph parameter, the projection theta number, and explores its properties and relation to existing theta variants, especially for vertex-transitive graphs.

Findings

Projection theta number is always less than or equal to Szegedy number.

Equality holds for vertex-transitive graphs.

The proposed method is computationally faster on vertex-transitive graphs.

Abstract

We introduce a new model for the chromatic number $\chi(G)$ based on what we call combinatorial projection matrices, which is a special class of doubly stochastic symmetric projection matrices. Relaxing this models yields an SDP whose optimal value is the projection theta number $\hat{\vartheta}(G)$, which is closely related to the Szegedy number $\vartheta^+(G)$, a variant of the Lov\'{a}sz theta number. We characterize that in general, $\hat{\vartheta}(G)\leq \vartheta^+(G)$, with equality if $G$ is vertex-transitive. While this seems to imply that working with binary matrices is a better paradigm than working with binary eigenvalues in this context, our approach is slightly faster than computing the Szegedy number on vertex-transitive graphs.