The non-cycle bound for the non-backtracking Laplacian parameter

Let $G$ be a graph with minimum degree

$$and let$$

denote the parameter introduced above. Assume that $G$ is not the cycle graph. Non-cycle bound.

$$\varepsilon\leq \frac{1}{\delta-1}.$$

This conjecture refines the preceding general bound for graphs of minimum degree at least two; its status is not determined by the supplied context.