Quasi-Irreducibility of Nonnegative Biquadratic Tensors

Liqun Qi, Chunfeng Cui () and Yi Xu
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Liqun Qi: Jiangsu Provincial Scientific Research Center of Applied Mathematics, Nanjing 211189, China
Chunfeng Cui: School of Mathematical Sciences, Beihang University, Beijing 100191, China
Yi Xu: Jiangsu Provincial Scientific Research Center of Applied Mathematics, Nanjing 211189, China

Mathematics, 2025, vol. 13, issue 13, 1-11

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.

Keywords: nonnegative biquadratic tensors; bipartite 2-graphs; quasi-irreducibility; M 0 -eigenvalues; M ++ -eigenvalues; max-min theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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