  The cyclic edge-connectivity of a graph $G$ is the least $k$ such that there
exists a set of $k$ edges whose removal disconnects $G$ into components where
every component contains a cycle. We show that for graphs of minimum degree at
least 3 and girth $g$ at least 4, the cyclic edge-connectivity is bounded above
by $(\Delta-2)g$ where $\Delta$ is the maximum degree. We then prove that if
the second eigenvalue of the adjacency matrix of a $d$-regular graph of girth
$g\geq4$ is sufficiently small, then the cyclic edge-connectivity is $(d-2)g$,
providing a spectral condition for when this upper bound on cyclic
edge-connectivity is tight.
