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 $S(2,k,n)$.

翻译：在张量特征值问题中，人们往往更关注张量的H-特征值。非负张量或一致超图的最大H-特征值即为该张量或一致超图的谱半径。本文研究了偶数阶或非负对称零对角张量的主子张量最大H-特征值的上界与下界（交错不等式），以及移除部分顶点或边的一致超图最大H-特征值的下界。同时，我们探讨了最小H-特征值的类似问题。通过实例验证了界的锐利性，并在某些一致超图情形中刻画了等式成立的条件。特别地，对于连通线性$k$一致超图$G$（其中$v\in V(G)$），我们给出了$G-v$谱半径关于$G$谱半径与顶点$v$度数的锐利下界，并刻画了极值超图；同时证明了移除一个顶点后子超图的最大谱半径不小于原超图谱半径减一，且等号成立当且仅当该超图为Steiner系统$S(2,k,n)$。