Adjacency Spectra of Random and Complete Hypergraphs

Abstract We present progress on the problem of asymptotically describing the adjacency eigenvalues of random and complete uniform hypergraphs. There is a natural conjecture arising from analogy with random matrix theory that connects these spectra to that of the all-ones hypermatrix. Several of the ingredients along a possible path to this conjecture are established, and may be of independent interest in spectral hypergraph/hypermatrix theory. In particular, we provide a bound on the spectral radius of the symmetric Bernoulli hyperensemble, and show that the spectrum of the complete k-uniform hypergraph for k = 2 , 3 is close to that of an appropriately scaled all-ones hypermatrix.