Spectral characterizations of graphs with small spectral radius

A graph is said to have a small spectral radius if it does not exceed the corresponding Hoffmann limit value. In the case of (signless) Laplacian matrix, the Hoffmann limit value is equal to ϵ+2=4.38+, with ϵ being the real root of x3-4x-4. Here the spectral characterization of connected graphs with small (signless) Laplacian spectral radius is considered. It is shown that all connected graphs with small Laplacian spectral radius are determined by their Laplacian spectra, and all but one of connected graphs with small signless Laplacian spectral radius are determined by their signless Laplacian spectra.