Spectral Threshold for Extremal Cyclic Edge-Connectivity

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