Lattices from tight frames and vertex transitive graphs

Lenny Fukshansky, Deanna Needell, Josiah Park, Yuxin Xin

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.

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