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Let $\lambda_{2}(G)$ be the second smallest normalized Laplacian eigenvalueof a graph $G$. In this paper, we determine all unicyclic graphs of order$n\geq21$ with $\lambda_{2}(G)\geq 1-\frac{\sqrt{6}}{3}$. Moreover, theunicyclic graphs with $\lambda_{2}(G)=1-\frac{\sqrt{6}}{3}$ are alsodetermined.

The unicyclic graphs with the second smallest normalized Laplacian eigenvalue no less than $1-\frac{\sqrt{6}}{3}$