Let G be a finite graph with H as a star complement for a non-zero eigenvalue μ. Let κ'(G), δ(G) denote respectively the edge-connectivity and minimum degree of G. We show that κ'(G) is controlled by δ(G) and κ'(H). We describe the possibilities for a minimum cutset of G when μ∉{-1,0}. For such μ, we establish a relation between κ'(G) and the spectrum of H when G has a non-trivial minimum cutset E⊈E(H).