Quasi-Irreducibility of Nonnegative Biquadratic Tensors

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.