Solving the problem about the second largest normalized Laplacian eigenvalue

Let G be a simple graph of order n with normalized Laplacian eigenvalue rho 1 >= rho 2 >= <middle dot> <middle dot> <middle dot> >= 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 = n-1 n-2 . (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.