Spectral Threshold for Extremal Cyclic Edge-Connectivity

The universal 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 universal 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 $$g\ge 4$$ is sufficiently small, then the universal cyclic edge-connectivity is $$(d-2)g$$ , providing a spectral condition for when this upper bound on universal cyclic edge-connectivity is tight.