Unlocking the walk matrix of a graph

Abstract

Let G be a graph on the vertex set V = {v 1 , . . . , v n } with adjacency matrix A. For a subset S of V let e = (x 1 , . . . , x n ) T be the characteristic vector of S, that is, x = 1 if v ∈ S and x = 0 otherwise. Then the walk matrix of G for S is

Flat bands by latent symmetry

“…( 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):…”

“…If the spectrum of H is degenerate or has any eigenvector with vanishing amplitudes on {u, v}, then W u and W v do not have full rank [47,52] and are thus not invertible. Hence, although a Q matrix still exists, which is unique under the convention of treating eigenvectors vanishing on {u, v} as {u, v}even [47], it cannot be obtained directly from Eq.…”