IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

The Spectrum of the All-Ones Hypermatrix

  • Feb. 28, 2013
  • 3:30 p.m.
  • LeConte 312

Abstract

Spectral graph theory relates the eigenvalues of matrices associated with graphs to properties of those graphs. Quite recently, a few different generalizations to (uniform) hypergraphs have been proposed. Here, we take the "algebraic" approach, using so-called resultants and symmetric hyperdeterminants to define the spectrum of a hypergraph's adjacency hypermatrix. In this talk, I will introduce the basic framework and then discuss recent results describing the spectrum of the all-ones hypermatrix, a surprisingly difficult task. This description includes a complete explanation of the eigenvalues' multiplicities, a so-far elusive aspect of the spectral theory of tensors. We also give a general distributional picture of the spectrum as a point-set in the complex plane and discuss possible implications for the spectrum of the complete hypergraph, about which there are a number of interesting questions. Joint work with Aaron Dutle of USC.

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