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Quasi-Irreducibility of Nonnegative Biquadratic Tensors

Liqun Qi, Chunfeng Cui, Yi Xu

TL;DR

The paper addresses spectral analysis of nonnegative biquadratic tensors, focusing on adjacency tensors of bipartite 2-graphs which are reducible, and introduces quasi-irreducibility to capture spectral properties. It defines $x$- and $y$-quasi-reducibility and proves that a non-bi-separable bipartite 2-graph yields a quasi-irreducible adjacency tensor, while the largest $M$-eigenvalue for quasi-irreducible tensors is constrained to be either an $M^0$- or an $M^{++}$-eigenvalue; furthermore, it establishes a max-min theorem for the $M$-spectral radius. The work also discusses the NP-hardness of computing the largest $M$-eigenvalue for general biquadratic tensors and delineates tractable cases within the nonnegative regime, including a proposed pathway via Problems 1–3 and related Collatz-type algorithms. Collectively, these results extend Perron-Frobenius-type spectral theory to quasi-irreducible nonnegative biquadratic tensors and inform spectral hypergraph analysis and algorithmic approaches for eigenvalue computation.

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.

Quasi-Irreducibility of Nonnegative Biquadratic Tensors

TL;DR

The paper addresses spectral analysis of nonnegative biquadratic tensors, focusing on adjacency tensors of bipartite 2-graphs which are reducible, and introduces quasi-irreducibility to capture spectral properties. It defines - and -quasi-reducibility and proves that a non-bi-separable bipartite 2-graph yields a quasi-irreducible adjacency tensor, while the largest -eigenvalue for quasi-irreducible tensors is constrained to be either an - or an -eigenvalue; furthermore, it establishes a max-min theorem for the -spectral radius. The work also discusses the NP-hardness of computing the largest -eigenvalue for general biquadratic tensors and delineates tractable cases within the nonnegative regime, including a proposed pathway via Problems 1–3 and related Collatz-type algorithms. Collectively, these results extend Perron-Frobenius-type spectral theory to quasi-irreducible nonnegative biquadratic tensors and inform spectral hypergraph analysis and algorithmic approaches for eigenvalue computation.

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-eigenvalues or M-eigenvalues. Furthermore, we establish a max-min theorem for the M-spectral radius of a nonnegative biquadratic tensor.

Paper Structure

This paper contains 6 sections, 8 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bipartite $2$-Graphs
  4. Quasi-Irreducible Nonnegative Biquadratic Tensors
  5. A Max-Min Theorem for Nonnegative Biquadratic Tensors
  6. The Largest M-eigenvalue of A Nonnegative Biquadratic Tensor

Key Result

Theorem 2.1

Suppose that $\mathcal{A} \in BQ(m, n)$. Then $\mathcal{A}$ always has M-eigenvalues. Furthermore, $\mathcal{A}$ is positive semi-definite if and only if all of its M-eigenvalues are nonnegative, and $\mathcal{A}$ is positive definite if and only if all of its M-eigenvalues are positive.

Figures (1)

  • Figure 1: Illustration of $x$-irreducible, $y$-irreducible, irreducible, and quasi-irreducible (i.e., both $x$- and $y$-quasi-irreducible) nonnegative biquadratic tensors for $m=n=2$.

Theorems & Definitions (14)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Theorem 4.3
  • ...and 4 more