Star complements and connectivity in finite graphs

Let G be a finite graph with H as a star complement for an eigenvalue other than 0 or -1. Let kappa(G), delta(G) denote respectively the vertex-connectivity and minimum degree of G. We prove that kappa(G) is controlled by delta(G) and kappa(H). In particular, for each k is an element of N there exists a smallest non-negative integer f(k) such that kappa(G) >= k whenever kappa(H) >= k and delta(G) >= f(k). We show that f (1) = 0, f (2) = 2, f (3) = 3, f (4) = 5 and f (5) = 7. (C) 2013 Elsevier Inc. All rights reserved.