Quasi-Irreducibility of Nonnegative Biquadratic Tensors

2025/04/14 by Qi, Liqun, Cui, Chunfeng, Xu, Yi · 2 citations
#FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2504.10129

Abstract

While the adjacency tensor of a bipartite 2-graph is a nonnegative biquadratic tensor, it is inherently reducible. To address this limitation, we introduce the concept of quasi-irreducibility in this paper. The adjacency tensor of a bipartite 2-graph is quasi-irreducible if that bipartite 2-graph is not bi-separable. This new concept reveals important spectral properties: although all M⁺-eigenvalues are M⁺⁺-eigenvalues for irreducible nonnegative biquadratic tensors, the M⁺-eigenvalues of a quasi-irreducible nonnegative biquadratic tensor can be either M⁰-eigenvalues or M⁺⁺-eigenvalues. Furthermore, we establish a max-min theorem for the M-spectral radius of a nonnegative biquadratic tensor.

Citations

• Nonnegative Biquadratic Tensors
• Biquadratic Tensors: Eigenvalues and Structured Tensors
• Mathematical Formulation of Multi-Layer Networks
• The Largest Laplacian and Signless Laplacian H-Eigenvalues of a Uniform Hypergraph
• Spectra of uniform hypergraphs
• Perron–Frobenius theorem for nonnegative multilinear forms and extensions

Related

• Nonnegative Biquadratic Tensors
• Biquadratic Tensors: Eigenvalues and Structured Tensors
• The Dimension of Eigenvariety of Nonnegative Tensors Associated with Spectral Radius
• Sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor and its applications
• Characterizations of the spectral radius of nonnegative weakly irreducible tensors via digraph