Some spectral properties and characterizations of connected odd-bipartite uniform hypergraphs

Abstract

A -uniform hypergraph is called odd-bipartite, if is even and there exists some proper subset of such that each edge of contains odd number of vertices in . Odd-bipartite hypergraphs are generalizations of the ordinary bipartite graphs. We study the spectral properties of the connected odd-bipartite hypergraphs. We prove that the Laplacian H-spectrum and signless Laplacian H-spectrum of a connected -uniform hypergraph are equal if and only if is even and is odd-bipartite. We further give several spectral characterizations of the connected odd-bipartite hypergraphs. We also give a characterization for a connected -uniform hypergraph whose Laplacian spectral radius and signless Laplacian spectral radius are equal; thus, provide an answer to a question raised by L. Qi. By showing that the Cartesian product of two odd-bipartite -uniform hypergraphs is still odd-bipartite, we determine that the Laplacian spectral radius of is the sum of the Laplacian spectral radii of and , when and are both connected odd-bipartite.