Hypergraphs and hypermatrices with symmetric spectrum

Abstract

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.

Eigenvectors of Laplacian or signless Laplacian of hypergraphs associated with zero eigenvalue

“…By the definition (2.1), (4) ⇔ (7). By Corollary 19 of [22], (1) ⇔ (8). So it suffices to prove (6) ⇒ (4) and (9) ⇒ (7).…”

“…Let G be hypergraph. A set U of vertices of G is called an odd (even) transversal if every edge of G intersects U in an odd (even) number of vertices [6,24,22]. G is called odd (even)-traversal if G has an odd (even) transversal.…”

“…An odd-bipartite hypergraph is odd-colorable, but the converse does not hold. Nikiforov [22] proved that if m ≡ 2 mod 4, then an m-uniform odd-colorable hypergraph is odd-bipartite. For m ≡ 0 mod 4, Nikiforov [22] constructed two families of m-uniform odd-colorable hypergraphs which are not odd-bipartite.…”

“…There are many results on the above tensors associated with hypergraph, see e.g. [5,7,8,13,14,15,17,18,19,21,22,23,27,29,33]. Shao et al [29] used the equality of H-spectra of the Laplacian and signless Laplacian tensor to characterize the oddbipartiteness of a hypergraph.…”

“…[22] Let G be an m-uniform hypergraph on n vertices, where m is even. The hypergraph G is called odd-colorable if there exists a map f : V (G) → [m] such that for each e ∈ E(G),…”