Sharp spectral bounds for the edge-connectivity of regular graphs

Let λ₂(G) and κ^′(G) be the second largest eigenvalue and the edge-connectivity of a graph G, respectively. Let r be a positive integer at least 3. For t=1 or 2, Cioabǎ gave sharp upper bounds for λ₂(G) in an r-regular simple graph G to guarantee that κ^′(G)≥t+1. In this paper, we resolve this question for all t≥3; if G is an r-regular simple graph with [Formula presented], then κ^′(G)≥t+1, and for odd t, if G is an r-regular simple graph with [Formula presented], then κ^′(G)≥t+1.