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 $G^{k,s}$, which is obtained from $G$ by blowing up each vertex into a $k$-set and each edge into a $(k-2s)$-set, where $s \le k/2$. When $s < k/2$, $G^{k,s}$ is always odd-bipartite. We show that $G^{k,{k \over 2}}$ is non-odd-bipartite if and only if $G$ is non-bipartite, and find that $G^{k,{k \over 2}}$ 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 $G^{k,{k \over 2}}$. In particular, we characterize the unique graph with minimum adjacency or signless Laplacian spectral radius among all non-odd-bipartite hypergraphs $G^{k,{k \over 2}}$ of fixed order, and prove that $\sqrt{2+\sqrt{5}}$ is the smallest limit point of the non-odd-bipartite hypergraphs $G^{k,{k \over 2}}$. In addition we obtain some results for the spectral radii of the weakly irreducible nonnegative tensors.