Star complements and edge-connectivity in finite graphs

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