Eigenvalues and edge-connectivity of regular graphs

AbstractIn this paper, we show that if the second largest eigenvalue of a d-regular graph is less than d-2(k-1)d+1, then the graph is k-edge-connected. When k is 2 or 3, we prove stronger results. Let ρ(d) denote the largest root of x3-(d-3)x2-(3d-2)x-2=0. We show that if the second largest eigenvalue of a d-regular graph G is less than ρ(d), then G is 2-edge-connected and we prove that if the second largest eigenvalue of G is less than d-3+(d+3)2-162, then G is 3-edge-connected