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 $G^{k,\frac{k}{2}}$ is non-odd-bipartite if and only if $G$ is non-bipartite, and find that $G^{k,\frac{k}{2}}$ 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 $G^{k,\frac{k}{2}}$. In particular, we characterize the unique graph with minimum adjacency or signless Laplacian spectral radius among all non-odd-bipartite hypergraphs $G^{k,\frac{k}{2}}$ of fixed order, and prove that $\sqrt{2+\sqrt{5}}$ is the smallest limit point of the non-odd-bipartite hypergraphs $G^{k,\frac{k}{2}}$. In addition we obtain some results for the spectral radii of the weakly irreducible nonnegative tensors.

References in corpus (3)

• Perron-Frobenius theorem for nonnegative multilinear forms and extensions
• On some properties of nonnegative weakly irreducible tensors
• Connected Hypergraphs with Small Spectral Radius

Cited by in corpus (19)

• The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs
• The $H$-spectrum of a generalized power hypergraph
• Eigenvectors of Laplacian or signless Laplacian of Hypergraphs Associated with Zero Eigenvalue
• The spectrum of a class of uniform hypergraphs
• The largest $H$-eigenvalue and spectral radius of Laplacian tensor of non-odd-bipartite generalized power hypergraphs
• The trace and Estrada index of uniform hypergraphs with cut vertices
• The cyclic index of adjacency tensor of generalized power hypergraphs
• Computing The Analytic Connectivity of A Uniform Hypergraph
• The spectral property of hypergraph coverings
• A spectral method to incidence balance of oriented hypergraphs and induced signed hypergraphs
• Some bounds on the eigenvalues of uniform hypergraphs
• The smallest spectral radius of bicyclic uniform hypergraphs with a given size
• The least H-eigenvalue of adjacency tensor of hypergraphs with cut vertices
• The least H-eigenvalue of signless Laplacian of non-odd-bipartite hypergraphs
• The effect on the spectral radius of r-graphs by grafting or contracting edges
• Minimal non-odd-transversal hypergraphs and minimal non-odd-bipartite hypergraphs
• Eigenvectors of Z-tensors associated with least H-eigenvalue with application to hypergraphs
• On the spectral radii of the unicyclic hypergraphs with fixed matching number
• The proof of a conjecture on largest Laplacian and signless Laplacian H-eigenvalues of uniform hypergraphs