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Quasi-Irreducibility of Nonnegative Biquadratic Tensors

Author

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  • Liqun Qi

    (Jiangsu Provincial Scientific Research Center of Applied Mathematics, Nanjing 211189, China
    Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong)

  • Chunfeng Cui

    (School of Mathematical Sciences, Beihang University, Beijing 100191, China)

  • Yi Xu

    (Jiangsu Provincial Scientific Research Center of Applied Mathematics, Nanjing 211189, China
    School of Mathematics, Southeast University, Nanjing 211189, China
    Nanjing Center for Applied Mathematics, Nanjing 211135, China)

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.

Suggested Citation

  • Liqun Qi & Chunfeng Cui & Yi Xu, 2025. "Quasi-Irreducibility of Nonnegative Biquadratic Tensors," Mathematics, MDPI, vol. 13(13), pages 1-10, June.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:13:p:2066-:d:1684689
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