“…( 2) beyond k = N − 1 since, by the Cayley-Hamilton theorem, any higher powers H k N can be expressed as lower order polynomials in H. As a consequence, if Eq. ( 2) holds for r = 0, .., N − 1, it automatically holds for all r. This enables the use of the N ×N walk matrix [51,52] W M of a subset M ⊆ H to encode walks ending in M, constructed by the action of H r on the indicator vector |e M of M (with m|e M = 1 for m ∈ M and 0 otherwise):…”