Edge-connectivity in regular multigraphs from eigenvalues

Let $G$ be a $d$-regular multigraph, and let ${\lambda }_2(G)$ be the second largest eigenvalue of $G$. In this paper, we prove that if $\lambda_2(G) < \frac{d-1+\sqrt{9d^2-10d+17}}4$,then$G$is2-edge-connected\mathrm{.}Furthermore,for$t\ge2$weshowthat$G$is$(t+1)$-edge-connectedwhen$\lambda_2(G)<d-t$, and in fact when ${\lambda }_2(G)<d-t+1$ if $t$ is odd.