Radial eigenvectors of the Laplacian of the nonbinary hypercube

Let q be a positive integer ⩾2. Define a (n+1)×(n+1) real, symmetric, tridiagonal matrix M, with rows and columns indexed by {0,1,…,n}, and with entries given by:M(i,j)=-(q-1)j(n-j+1)ifi=j-1,j+(q-1)(n-j)ifi=j,-(q-1)(j+1)(n-j)ifi=j+1,0if|i-j|⩾2.The (n+1)-dimensional space of radial vectors for the nonbinary hypercube Cq(n) is invariant under the Laplacian and M arises as the matrix of the Laplacian with respect to a suitable orthonormal basis of radial vectors. We show that the eigenvalues of M are 0,q,2q,…,nq by explicitly writing down the eigenvectors.