Some graphs determined by their spectra

Let denote the graph obtained by attaching m pendent edges to a vertex of complete graph Kn-m, and Un,p the graph obtained by attaching n-p pendent edges to a vertex of Cp. In this paper, we first prove that the graph and its complement are determined by their adjacency spectra, and by their Laplacian spectra. Then we prove that Un,p is determined by its Laplacian spectrum, as well as its adjacency spectrum if p is odd, and find all its cospectral graphs for Un,4.