Some graphs determined by their spectra

Let denote the graph obtained by attaching m pendent edges to a vertex of complete graph K_n-m, and U_n,p the graph obtained by attaching n-p pendent edges to a vertex of C_p. 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 U_n,p is determined by its Laplacian spectrum, as well as its adjacency spectrum if p is odd, and find all its cospectral graphs for U_n,4.