Let K_m is a complete graph of order m. By linking n+1 complete graphs of order m in a fixed way, a complete associated graph H_n,Km of order (mn + m) was obtained. The adjacency eigenvalues, Laplacian eigenvalues and signless Laplacian eigenvalues of H_n,Km were provided according to the relevant conclusions about quotient matrix and rank. So the adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum of H_n,Km were determined. At the same time, based on the study of Brualdi-Solheid spectral radius, the research of this kind of spectral radius problem was extended to the study of Laplacian spectral radius and signless Laplacian spectral radius of graphs. The upper and lower bounds of the radius of adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum of H_n,Km (the set of complete correlated graphs with points N where N=m(n + 1)) were provided. And the extremal graphs were described where the radius of adjacency spectrum attained the upper bound and Laplacian spectrum and signless Laplacian spectrum attained the upper or lower bounds.