# Graph Theory Data for Topological Quantum Chemistry

## TL;DR

This paper develops algorithms and provides data to classify all topologically distinct band structures in materials using graph theory, bridging local symmetry data with global topological properties.

## Contribution

It introduces explicit data and algorithms to generate all topologically distinct graphs representing band structures, enabling systematic classification of topological phases.

## Key findings

- Tabulated all orbital types and lattices leading to topologically disconnected bands
- Developed algorithms to produce all topologically distinct graphs
- Integrated results into BANDREP on Bilbao Crystallographic Server

## Abstract

Topological phases of noninteracting particles are distinguished by global properties of their band structure and eigenfunctions in momentum space. On the other hand, group theory as conventionally applied to solid-state physics focuses only on properties which are local (at high symmetry points, lines, and planes) in the Brillouin zone. To bridge this gap, we have previously [B. Bradlyn et al., Nature 547, 298--305 (2017)] mapped the problem of constructing global band structures out of local data to a graph construction problem. In this paper, we provide the explicit data and formulate the necessary algorithms to produce all topologically distinct graphs. Furthermore, we show how to apply these algorithms to certain "elementary" band structures highlighted in the aforementioned reference, and so identified and tabulated all orbital types and lattices that can give rise to topologically disconnected band structures. Finally, we show how to use the newly developed BANDREP program on the Bilbao Crystallographic Server to access the results of our computation.