# The Theta Number of Simplicial Complexes

## TL;DR

This paper generalizes the Lovász theta number from graphs to simplicial complexes using cohomology and Laplacians, providing new bounds on independence numbers through semidefinite programming.

## Contribution

It introduces a novel higher-dimensional theta number for simplicial complexes based on cohomology and Laplacians, extending graph bounds to complex structures.

## Key findings

- The higher-dimensional theta number relates to Hoffman's ratio bound and chromatic number.
- A hierarchy of semidefinite bounds converges to the independence number.
- Analysis of the theta number on dense random complexes shows its effectiveness.

## Abstract

We introduce a generalization of the celebrated Lov\'asz theta number of a graph to simplicial complexes of arbitrary dimension. Our generalization takes advantage of real simplicial cohomology theory, in particular combinatorial Laplacians, and provides a semidefinite programming upper bound of the independence number of a simplicial complex. We consider properties of the graph theta number such as the relationship to Hoffman's ratio bound and to the chromatic number and study how they extend to higher dimensions. Like in the case of graphs, the higher dimensional theta number can be extended to a hierarchy of semidefinite programming upper bounds reaching the independence number. We analyze the value of the theta number and of the hierarchy for dense random simplicial complexes.