Spectral characterization of graphs whose second largest eigenvalue is less than 1

Graphs with second largest eigenvalue λ2⩽1 are extensively studied, however, whether they are determined by their adjacency spectra or not is less considered. In this paper we completely characterize all the connected bipartite graphs with λ2<1 that are determined by their adjacency spectra. In addition, we prove that all the connected non-bipartite graphs with girth no less than 4 and λ2<1 are determined by their adjacency spectra.