Solving the problem about the second largest normalized Laplacian eigenvalue

Abstract

Let $G$ be a simple graph of order $n$ with normalized Laplacian eigenvalue ${\rho }_1\ge {\rho }_2\ge \cdots \ge {\rho }_n=0$. In Sun and Das (2019), the authors obtained the first three smallest values on ${\rho }_2$ of connected graphs and proposed an open problem on the third smallest values of ${\rho }_2$. In this paper, we solve the problem completely, characterize all connected graphs with order $n$, which satisfy ${\rho }_2=\frac{n-1}{n-2}$.