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 $ngeq21$ with $lambda_{2}(G)geq 1-frac{sqrt{6}}{3}$. Moreover, the unicyclic graphs with $lambda_{2}(G)=1-frac{sqrt{6}}{3}$ are also determined.