Largest and Least H-Eigenvalues of Symmetric Tensors and Hypergraphs

TL;DR

This paper investigates bounds for the largest and least H-eigenvalues of symmetric tensors and hypergraphs, providing new inequalities, characterizations, and extremal cases for spectral radii and eigenvalues.

Contribution

It introduces new bounds and inequalities for H-eigenvalues of symmetric tensors and hypergraphs, including sharp bounds and characterizations of extremal structures.

Findings

Established bounds for the largest H-eigenvalue of principal subtensors.

Derived lower bounds for the spectral radius of hypergraphs with vertices removed.

Characterized extremal hypergraphs achieving equality in bounds.

Abstract

In tensor eigenvalue problems, one is likely to be more interested in H-eigenvalues of tensors. The largest H-eigenvalue of a nonnegative tensor or of a uniform hypergraph is the spectral radius of the tensor or of the uniform hypergraph. We find upper bounds and lower bounds (interlacing inequalities) for the largest H-eigenvalue of a principal subtensor of a symmetric zero diagonal tensor that is of even order or nonnegative, as well as lower bounds for the largest H-eigenvalue of a uniform hypergraph with some vertices or edges removed. We also investigate similar problems for the least H-eigenvalues. We give examples to verify the sharpness of the bounds or in some cases for uniform hypergraphs, we characterize the equality. Particularly, for a connected linear $k$-uniform hypergraph $G$ with $v\in V(G)$, we give a sharp lower bound for the spectral radius of $G-v$ in terms of the spectral radius of $G$ and the degree of $v$ and characterize the extremal hypergraphs, and show that the maximum spectral radius of the subhypergraphs with one vertex removed is greater than or equal to the spectral radius of the hypergraph minus one, which is attained if and only if it is a Steiner system $S(2,k,n)$.