On the spectral radius of a class of non-odd-bipartite even uniform hypergraphs

Abstract

In order to investigate the non-odd-bipartiteness of even uniform hypergraphs, starting from a simple graph G, we construct a generalized power of G, denoted by Gk,s, which is obtained from G by blowing up each vertex into a s-set and each edge into a (k−2s)-set, where s≤k/2. When s<k/2, Gk,s is always odd-bipartite. We show that Gk,k2 is non-odd-bipartite if and only if G is non-bipartite, and find that Gk,k2 has the same adjacency (respectively, signless Laplacian) spectral radius as G. So the results involving the adjacency or signless Laplacian spectral radius of a simple graph G hold for Gk,k2. In particular, we characterize the unique graph with minimum adjacency or signless Laplacian spectral radius among all non-odd-bipartite hypergraphs Gk,k2 of fixed order, and prove that 2+5 is the smallest limit point of the non-odd-bipartite hypergraphs Gk,k2. In addition we obtain some results for the spectral radii of the weakly irreducible nonnegative tensors.