# Lattices from tight frames and vertex transitive graphs

## TL;DR

This paper establishes a link between tight frames, lattices, and vertex transitive graphs, showing how certain symmetric graphs generate well-structured lattices with notable properties.

## Contribution

It introduces a construction method for lattices from vertex transitive graphs and characterizes their properties, including strong eutaxy in the case of irreducible group frames.

## Key findings

- Lattices from irreducible group frames are strongly eutactic
- Existence of such lattices in arbitrarily large dimensions
- Recovery of well-known root lattices through this construction

## Abstract

We show that real tight frames that generate lattices must be rational, and use this observation to describe a construction of lattices from vertex transitive graphs. In the case of irreducible group frames, we show that the corresponding lattice is always strongly eutactic. This is the case for the more restrictive class of distance transitive graphs. We show that such lattices exist in arbitrarily large dimensions and demonstrate examples arising from some notable families of graphs. In particular, some well-known root lattices and those related to them can be recovered this way. We discuss various properties of this construction and also mention some potential applications of lattices generated by incoherent systems of vectors.