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 κ(G), δ(G) denote respectively the vertex-connectivity and minimum degree of G. We prove that κ(G) is controlled by δ(G) and κ(H). In particular, for each k ∈ IN there exists a smallest non-negative integer f(k) such that κ(G) ≥ k whenever κ(H) ≥ k and δ(G) ≥ f(k). We show that f(1) = 0, f(2) = 2, f(3) = 3, f(4) = 5 and f(5) = 7. AMS Classification: 05C50