Two New Eigenvalue Inclusion Sets for Sparse Tensors and Their Applications

Sparse tensors have been widely applied in hypergraph representation, complex network analysis, and high-dimensional data processing, where eigenvalue inclusion sets play a fundamental role in analyzing the properties of such tensors. For sparse tensors, two new eigenvalue inclusion sets are proposed in this paper via the matrix’s digraph. Specifically, we employ the digraphs of the majorization matrix and the representation matrix to establish two new S-type inclusion sets and theoretically prove that these sets are tighter than the existing ones. By applying these inclusion sets, some inequalities for the spectral radius of weakly irreducible nonnegative tensors are derived. Moreover, several criteria for verifying the positive (semi-)definiteness of symmetric sparse tensors are also obtained. Numerical examples are provided to illustrate the effectiveness of the proposed spectral radius bounds and positive definiteness criteria.