Quasi-Irreducibility of Nonnegative Biquadratic Tensors
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...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-06-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/13/2066 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850118876133588992 |
|---|---|
| author | Liqun Qi Chunfeng Cui Yi Xu |
| author_facet | Liqun Qi Chunfeng Cui Yi Xu |
| author_sort | Liqun Qi |
| collection | DOAJ |
| description | 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<sup>+</sup>-eigenvalues are M<sup>++</sup>-eigenvalues for irreducible nonnegative biquadratic tensors, the M<sup>+</sup>-eigenvalues of a quasi-irreducible nonnegative biquadratic tensor can be either M<sup>0</sup>-eigenvalues or M<sup>++</sup>-eigenvalues. Furthermore, we establish a max-min theorem for the M-spectral radius of a nonnegative biquadratic tensor. |
| format | Article |
| id | doaj-art-e9aee2f047f547629ef2836da49badf4 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-e9aee2f047f547629ef2836da49badf42025-08-20T02:35:46ZengMDPI AGMathematics2227-73902025-06-011313206610.3390/math13132066Quasi-Irreducibility of Nonnegative Biquadratic TensorsLiqun Qi0Chunfeng Cui1Yi Xu2Jiangsu Provincial Scientific Research Center of Applied Mathematics, Nanjing 211189, ChinaSchool of Mathematical Sciences, Beihang University, Beijing 100191, ChinaJiangsu Provincial Scientific Research Center of Applied Mathematics, Nanjing 211189, ChinaWhile 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<sup>+</sup>-eigenvalues are M<sup>++</sup>-eigenvalues for irreducible nonnegative biquadratic tensors, the M<sup>+</sup>-eigenvalues of a quasi-irreducible nonnegative biquadratic tensor can be either M<sup>0</sup>-eigenvalues or M<sup>++</sup>-eigenvalues. Furthermore, we establish a max-min theorem for the M-spectral radius of a nonnegative biquadratic tensor.https://www.mdpi.com/2227-7390/13/13/2066nonnegative biquadratic tensorsbipartite 2-graphsquasi-irreducibilityM<sup>0</sup>-eigenvaluesM<sup>++</sup>-eigenvaluesmax-min theorem |
| spellingShingle | Liqun Qi Chunfeng Cui Yi Xu Quasi-Irreducibility of Nonnegative Biquadratic Tensors Mathematics nonnegative biquadratic tensors bipartite 2-graphs quasi-irreducibility M<sup>0</sup>-eigenvalues M<sup>++</sup>-eigenvalues max-min theorem |
| title | Quasi-Irreducibility of Nonnegative Biquadratic Tensors |
| title_full | Quasi-Irreducibility of Nonnegative Biquadratic Tensors |
| title_fullStr | Quasi-Irreducibility of Nonnegative Biquadratic Tensors |
| title_full_unstemmed | Quasi-Irreducibility of Nonnegative Biquadratic Tensors |
| title_short | Quasi-Irreducibility of Nonnegative Biquadratic Tensors |
| title_sort | quasi irreducibility of nonnegative biquadratic tensors |
| topic | nonnegative biquadratic tensors bipartite 2-graphs quasi-irreducibility M<sup>0</sup>-eigenvalues M<sup>++</sup>-eigenvalues max-min theorem |
| url | https://www.mdpi.com/2227-7390/13/13/2066 |
| work_keys_str_mv | AT liqunqi quasiirreducibilityofnonnegativebiquadratictensors AT chunfengcui quasiirreducibilityofnonnegativebiquadratictensors AT yixu quasiirreducibilityofnonnegativebiquadratictensors |