The largest H-eigenvalue and spectral radius of Laplacian tensor of non-odd-bipartite generalized power hypergraphs.

Let G be a simple graph or hypergraph, and let A ( G ) , L ( G ) , Q ( G ) be the adjacency, Laplacian and signless Laplacian tensors of G respectively. The largest H -eigenvalues (respectively, the spectral radii) of L ( G ) , Q ( G ) are denoted respectively by λ max L ( G ) , λ max Q ( G ) (respectively, ρ L ( G ) , ρ Q ( G ) ). It is known that for a connected non-bipartite simple graph G , λ max L ( G ) = ρ L ( G ) < ρ Q ( G ) . But this does not hold for non-odd-bipartite hypergraphs. We will investigate this problem by considering a class of generalized power hypergraphs G k , k 2 , which are constructed from simple connected graphs G by blowing up each vertex of G into a k 2 -set and preserving the adjacency of vertices. Suppose that G is non-bipartite, or equivalently G k , k 2 is non-odd-bipartite. We get the following spectral properties: (1) ρ L ( G k , k 2 ) = ρ Q ( G k , k 2 ) if and only if k is a multiple of 4; in this case λ max L ( G k , k 2 ) < ρ L ( G k , k 2 ) . (2) If k ≡ 2 ( mod 4 ) , then for sufficiently large k , λ max L ( G k , k 2 ) < ρ L ( G k , k 2 ) . Motivated by the study of hypergraphs G k , k 2 , for a connected non-odd-bipartite hypergraph G , we give a characterization of L ( G ) and Q ( G ) having the same spectra or the spectrum of A ( G ) being symmetric with respect to the origin, that is, L ( G ) and Q ( G ) , or A ( G ) and − A ( G ) are similar via a complex (necessarily non-real) diagonal matrix with modular-1 diagonal entries. So we give an answer to a question raised by Shao et al., that is, for a non-odd-bipartite hypergraph G , that L ( G ) and Q ( G ) have the same spectra can not imply they have the same H -spectra. [ABSTRACT FROM AUTHOR]