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

Let ${\lambda }_2(G)$ be the second smallest normalized Laplacian eigenvalue of a graph $G$. In this paper, we determine all unicyclic graphs of order $n\ge 21$ with ${\lambda }_2(G)\ge 1-\frac{\sqrt{6}}{3}$. Moreover, the unicyclic graphs with ${\lambda }_2(G)=1-\frac{\sqrt{6}}{3}$ are also determined.