Let \((M,g)\) be a compact oriented Riemannian manifold without boundary, of dimension \(n\), \(\text{spec}^p (M)\) the non-decreasing sequence of the eigenvalues \(\lambda_p\) for the Laplacian acting on \(p\)-forms \((p=0,1,2, \dots,n)\). The author investigates the relation between some minimal submanifolds of the unit sphere \(S^{n+1}(1)\) and their spectra and proves that, if \(M\) is a minimal submanifold of \(S^{n+1} (1)\) and \(M'=M_{m,n-m} =S^m (\sqrt m/n) \times S^{n-m} (\sqrt {n-m}/n)\) is the Clifford minimal hypersuperface of \(S^{n+ 1} (1)\), and if \(\text{spec}^1(M) =\text{spec}^1(M')\), then, under some conditions on the curvature and on \(n\), \(M\) coincides with \(M'\).