Quasi-Irreducibility of Nonnegative Biquadratic Tensors

Liqun Qi; Chunfeng Cui; Yi Xu

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arXiv · 2504.10129

Quasi-Irreducibility of Nonnegative Biquadratic Tensors

Liqun Qi·Chunfeng Cui·Yi Xu

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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 M$^0$-eigenvalues or M$^{++}$-eigenvalues. Furthermore, we establish a max-min theorem for the M-spectral radius of a nonnegative biquadratic tensor.

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BibTeX RIS

Liqun Qi, Chunfeng Cui, Yi Xu. 2025-04-14. Quasi-Irreducibility of Nonnegative Biquadratic Tensors. https://arxiv.org/abs/2504.10129

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