On the construction of equiangular frames from graphs

We give details of the 1-1 correspondence between equiangular frames of n vectors for Rd 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 openR3. Further, the associated canonical tight frames have a small number of angles in many cases.