Spectra of graphs attached to the space of melodies

For any integer n greater than or equal to two, two intimately related graphs on the vertices of the n-dimensional cube are introduced. All of their eigenvalues are found to be integers, and the largest and the smallest ones are also determined. As a byproduct, certain kind of generating function for their spectra is introduced and shown to be quite effective to compute the eigenvalues of some broader class of adjacency matrices of graphs.

► Two graphs on the hypercube are introduced. ► All of their eigenvalues are found to be integers. ► Their largest and smallest eigenvalues are determined. ► Crucial role is played by a generating function for the spectra of each graph. ► A possible generalization to some other graphs on the hypercube is indicated.