On least eigenvalues and least eigenvectors of real symmetric matrices and graphs

We give an upper bound for the least eigenvalue of a principal submatrix of a real symmetric matrix with zero diagonal, from which we establish an upper bound for the least eigenvalue of a graph when some vertices are removed using the components of the least eigenvector(s). We give lower and upper bounds for the least eigenvalue of a graph when some edges are removed. We also establish bounds for the components of the least eigenvector(s) of a real symmetric matrix and a graph.