Spectral characterization of unicyclic graphs whose second largest eigenvalue does not exceed 1

Connected graphs with the same number of vertices and edges are called unicyclic graphs. All unicyclic graphs whose second largest eigenvalue does not exceed 1, denoted by U(λ2), are determined by Xu [10]. Let Gt,s denote the graph obtained by attaching t≥1 pendent paths of length 2 and s≥0 pendent edges to a vertex of a triangle. In this paper, we first prove that the graph Gt,s is determined by its adjacency spectrum. Then we prove that the graphs in U(λ2) are determined by their adjacency spectra except for four graphs.