On the construction of equiangular frames from graphs

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Abstract

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.

AMS classification

Primary 42C15, 05C50
Secondary 05C90, 52B15

Keywords

Finite frame
Tight frame
Grassmannian frame
Mutually unbiased basis
Two angle frame
Seidel matrix
Adjacency matrix
Algebraic graph theory
Signal processing
Information theory

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