Eigenvalues of Non-Regular Linear-Quasirandom Hypergraphs

Chung, Graham, and Wilson proved that a graph is quasirandom if and only if there is a large gap between its first and second largest eigenvalue. Recently, the authors extended this characterization to k-uniform hypergraphs, but only for the so-called coregular k-uniform hypergraphs. In this paper, we extend this characterization to all k-uniform hypergraphs, not just the coregular ones. Specifically, we prove that if a k-uniform hypergraph satisfies the correct count of a specially defined four-cycle, then there is a gap between its first and second largest eigenvalue.Comment: 15 pages. (this paper was originally part of an old version of arXiv:1208.4863

Spectral Properties of Oriented Hypergraphs

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. The adjacency and Laplacian eigenvalues of an oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and Laplacian matrices of an oriented hypergraph which depend on structural parameters of the oriented hypergraph are found. An oriented hypergraph and its incidence dual are shown to have the same nonzero Laplacian eigenvalues. A family of oriented hypergraphs with uniformally labeled incidences is also studied. This family provides a hypergraphic generalization of the signless Laplacian of a graph and also suggests a natural way to define the adjacency and Laplacian matrices of a hypergraph. Some results presented generalize both graph and signed graph results to a hypergraphic setting.Comment: For the published version of the article see http://repository.uwyo.edu/ela/vol27/iss1/24

Hypergraphs and hypermatrices with symmetric spectrum

It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to hypermatrices. To this end the concept of bipartiteness is generalized by a new monotone property of cubical hypermatrices, called odd-colorable matrices. It is shown that a nonnegative symmetric $r$-matrix $A$ has a symmetric spectrum if and only if $r$ is even and $A$ is odd-colorable. This result also solves a problem of Pearson and Zhang about hypergraphs with symmetric spectrum and disproves a conjecture of Zhou, Sun, Wang, and Bu. Separately, similar results are obtained for the $H$-spectram of hypermatrices.Comment: 17 pages. Corrected proof on p. 1