Graphs with small second largest Laplacian eigenvalue

Let L ( G ) be the Laplacian matrix of G . In this paper, we characterize all of the connected graphs with second largest Laplacian eigenvalue no more than l ; where l ≐ 3.2470 is the largest root of the equation μ 3 − 5 μ 2 + 6 μ − 1 = 0 . Moreover, this result is used to characterize all connected graphs with second largest Laplacian eigenvalue no more than three.