Lower bounds for the smallest singular value

Let A = (aij) be an n × n complex matrix. Suppose that G(A), the undirected graph of A, has no isolated vertex. Let E be the set of edges of G(A). We prove that the smallest singular value of A, σn, satisfies: σn ≥ min σij (i, j) E, where gij ≡ ai + aj − [(ai − aj)2 + (ri + ci)(rj + cj)]1/2/2 with ai ≡ aii and ri,ci are the ith deleted absolute row sum and column sum of A, respectively. The result simplifies and improves that of Johnson and Szulc: σn ≥ mini≠j σij. (See [1].)