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 λmaxL(G), λmaxQ(G) (respectively, ρL(G), ρQ(G)). It is known that for a connected non-bipartite simple graph G, λmaxL(G)=ρL(G)<ρQ(G). But this does not hold for non-odd-bipartite hypergraphs. We will investigate this problem by considering a class of generalized power hypergraphs Gk,k2, which are constructed from simple connected graphs G by blowing up each vertex of G into a k2-set and preserving the adjacency of vertices.Suppose that G is non-bipartite, or equivalently Gk,k2 is non-odd-bipartite. We get the following spectral properties: (1) ρL(Gk,k2)=ρQ(Gk,k2) if and only if k is a multiple of 4; in this case λmaxL(Gk,k2)<ρL(Gk,k2). (2) If k≡2(mod4), then for sufficiently large k, λmaxL(Gk,k2)<ρL(Gk,k2). Motivated by the study of hypergraphs Gk,k2, 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.