# On the construction of equiangular frames from graphs

## Shayne Waldron

## Abstract:

We give details of the 1--1 correspondence between equiangular frames
of n vectors for R^d and graphs with n vertices.
This has been studied recently for *tight* equiangular frames
because of applications to signal processing and quantum information theory.
The *nontight* examples given here (which correspond to graphs with
more than 2 eigenvalues) have the potential for similar
applications, e.g., the frame corresponding to the 5-cycle graph is
the unique Grassmannian frame of 5 vectors in R^3.
Further, the associated canonical tight frames have
a *small* number of angles in many cases.

**Keywords:**
finite frame,
tight frame,
Grassmannian frame,
mutually unbiased basis,
two angle frame,
Seidel matrix,
adjacency matrix,
algebraic graph theory,
signal processing,
information theory,

**Math Review Classification:**
Primary 42C15, 05C50;
Secondary 05C90, 52B15

**Length:** 17 pages

**Last Updated:** 22 January 2009

## Availability: