Quasi-Irreducibility of Nonnegative Biquadratic Tensors
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910910854987776 |
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| author | Qi, Liqun Cui, Chunfeng Xu, Yi |
| author_facet | Qi, Liqun Cui, Chunfeng Xu, Yi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_10129 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi-Irreducibility of Nonnegative Biquadratic Tensors Qi, Liqun Cui, Chunfeng Xu, Yi Spectral Theory 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. |
| title | Quasi-Irreducibility of Nonnegative Biquadratic Tensors |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2504.10129 |